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IN  MEMORIAM 
FLORIAN  CAJORI 


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SCHOOL   ALGEBRA. 


BY 

C.   A.   VAN  VELZER, 

AND 

CHAS.   S.   SLIGHTER, 

PROFESSORS    IN    THE    UNIVERSITY   OF   WISCONSIN. 


MADISON,  WIS.: 

TRACY,  GIBBS  &  CO. 


COPYRIGHT, 

C.  A.  VAN  VELZER,  CHAS.  S.  SLIGHTER, 

1890. 


Tracy,  Gibbs  &  Co.,  Printers  and  Stereotypers. 


PREFACE. 


The  present  volume  is  icsued  in  the  hope  that  it  will  assist  the 
teacher  in  the  effort  to  get  pupils  to  think  and  that  it  will  induce  pupils 
to  place  the  subject  of  Algebra  on  a  rational  instead  of  an  arbitrary 
basis,  to  work  from  principles  rather  than  from  rules. 

In  the  first  part  of  the  work  we  have  pursued  what  is  termed  the 
inductive  method,  but  we  do  not  wish  this  understood  as  that  method 
which  infers  general  principles  from  an  accumulation  of  particular 
cases.  This  is  the  inductive  reasoning  of  the  natural  sciences,  but 
we  believe  it  is  never  legitimate  in  mathematics.  Induction,  as  we 
use  the  term,  means  that  method  which  proceeds  from  the  particular 
to  the  general.  By  particular  cases,  which  gradually  increase  in 
generality,  the  mind  of  the  learner  is  prepared  to  appreciate  the  gen- 
eral case,  but  this  general  case  must  so  present  itself  to  the  learner's 
mind  that  he  sees  that  the  truth  stated  must  be  so  and  cannot  possibly 
be  otherwise. 

It  will  be  noticed  that  we  have  not  thought  it  necessary  to  complete 
one  subject  before  taking  up  another,  but  subjects  have  sometimes 
been  treated  in  an  elementary  way  at  first  and  more  completely  at 
some  subsequent  part  of  the  book.  We  believe  that  by  this  plan 
students  can  follow  the  work  more  easily  and  with  more  profit,  and  at 
the  same  time  we  are  enabled  to  treat  some  subjects,  especially 
Factors,  Multiples,  and  Fractions,  more  fully  than  is  ordinarily  done. 

We  have  placed  the  principles  governing  the  use  of  parentheses 
before  the  four  fundamental  operations  of  addition,  subtraction,  mul- 
tiplication, and  division,  and  have  made  the  latter  depend  upon  the 
former.  This  we  think  enables  us  to  treat  the  four  fundamental 
operations  in  a  way  which  is  more  rational  to  beginners  than  is  given 
when  the  usual  order  is  pursued. 


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IV  PREFACE. 

The  subject  of  equations  is  early  introduced,  and  is  distributed 
through  the  book  instead  of  being  given  together  in  one  place.  This 
keeps  up  the  interest  in  the  subject  and  prevents  the  student  from 
getting  the  idea  that  he  is  learning  a  mass  of  theory  which  has  no 
practical  application. 

Indices  and  Surds  appear  after  Quadratics  for  the  reason  that  these 
subjects  are  more  difficult  than  Quadratics. 

Additive  and  subtractive  terms  are  distinguished  from  positive  and 
negative  quantities,  and  the  latter  are  postponed  until  after  the  four 
fundamental  operations. 

This  work  contains  about  3000  examples  besides  several  hundred 
problems  and  inductive  exercises.  Many  of  these  are  original  and 
many  are  taken  from  the  German  and  French  collections  and  from 
the  English  examination  papers.  The  answers  are  not  printed  in  the 
book  for  a  reason  that  every  teacher  of  Algebra  can  readily  assign,  but 
the  answers  are  issued  in  pamphlet  form  for  the  use  of  teachers  only. 

C.  A.  Van  Velzer. 
University  of  Wisconsin,  Chas.  S.  Slighter. 

December,  1890. 


TABLE  OF  CONTENTS. 


Bold  face  figures  refer  to  the  exercises  and 
light  face  figures  to  the  pages. 

CHAPTER   I. 
First  Principles,     .-----. 


1,  Illustrating  how  a  letter  may  be  used  to  represent  a 
number,  i.  2,  Leading  to  the  idea  of  an  algebraic  ex- 
pression, 3.  3,  Leading  to  the  idea  of  an  algebraic  equa- 
tion and  to  the  distinction  between  known  and  unknown 
numbers,  4.  4,  Symbolic  expressions,  6.  5,  Problems,  7. 
6,  Expressions  in  which  several  letters  arc  used  to  stand 
for  numbers,  10. 

CHAPTER  II. 

Union  of  Terms  and  Removal  of  Parentheses, 


15 


7,  Union  of  similar  terms,  15.  8,  Examples,  19.  9,  Paren- 
theses and  their  removal,  21.  lO,  Removal  of  parentheses; 
general  form  a-\-[b-\-c'),  22  ;  general  form  a-\-{b—c)^  22  ;  gen- 
eral form  a—{b-\-c\  23  ;  general  form  a  —  [b—c),  24  ;  paren- 
theses preceded  by  plus  sign,  24  ,  parentheses  preceded  by 
minus  sign,  26.  11,  Miscellaneous  examples  on  the  removal 
of  parentheses,  29. 

CHAPTER  III. 
Addition, ----30 

12,  Addition  of  expressions,  30;  examples,  31.  13,  Arrange- 
ment of  work  in  addition,  31.     14,  Examples,  33. 

CHAPTER   IV. 
Subtraction,    - -jg 

15,  Subtraction  of  expressions,  36;  examples,  37.  16,  Ar- 
rangement of  work  in  subtraction,  38.  17,  Examples,  38. 
1'8,  Addition  and  subtraction  of  equals,  40.  19,  Examples, 
42.  20,  Transposition  in  equations,  43.  21,  Examples,  44. 
22,  Problems,  46. 


vi  CONTENTS. 

CHAPTER  V. 
Multiplication,       _._------5o 

23,  General  definition  of  multiplication,  50 ;  examples,  51. 

24,  Multiplication  of  monomials,  52.  25,  Examples,  53. 
26,  Law  of  exponents  in  multiplication,  54.  27,  Examples, 
55.  28,  Multiplication  of  polynomials  by  monomials,  56. 
29,  Examples,  59.  30,  Arrangement  of  work  in  the  mul- 
tiplication of  a  polynomial  by  a  monomial,  59;  examples,  60. 
31,  Multiplication  of  polynomials  by  polynomials,  62.  32, 
Examples,  63.  33,  Arrangement  of  work  in  the  multipli- 
cation of  two  polynomials,  64.  34-,  Examples,  69.  35, 
Equations  involving  multiplication,  71. 

CHAPTER  VI. 
Division,  -- -        •        -        Ti 

36,  Division  of  monomials  by  monomials,  73  ;  the  law  of 
exponents,  74.  37,  Examples,  74.  38,  Division  of  poly- 
nomials by  monomials,  76.  39,  Examples,  77.  40,  Di- 
vision of  polynomials  by  polynomials,  77.  41,  Examples, 
79.  42,  Arrangement  of  work  in  division  of  polynomials 
by  polynomials,  81.     43,  Examples,  85. 

CHAPTER  Vn. 

Negative  Quantities, ---88 

44,  Number  and  quantity,  88.  45,  Opposite  directions, 
89.  46,  How  directions  are  distinguished,  90.  47,  Posi- 
tive and  negative  numbers,  92.  48,  Illustrative  examples, 
94.  49,  Addition,  97.  50,  Subtraction,  99.  51,  Multi- 
plication, 100.     52,  Division,  loi. 

CHAPTER  VIII. 
Parentheses,    -        - 103 

53,  Removal  of  parentheses,  103.  54,  Insertion  of  paren- 
theses, 104. 

CHAPTER  IX. 
Elementary  Factors,  Multiples,  and  Fractions,  -  -  106 
55,  Factors,  106.  56,  Highest  common  factor,  107.  57, 
Lowest  common  multiple,  108.  58,  Fractions,  iii.  59, 
Addition  of  fractions,  113;  subtraction  of  fractions,  115. 
60,  Multiplication  of  fractions,  116.  61,  Division  of  frac- 
tions, 118. 


CONTENTS.  vii 

CHAPTER  X. 
Simple  Equations, --       121 

62,  Definitions  and  general  principles,  121.  63,  Exam- 
ples, 126.  64,  Literal  equations,  127.  64-a,  Symbolic 
expressions,  129.     64&,  Problems,  132. 

CHAPTER  XI. 

Simultaneous  Equations, -      139 

65,  Definitions  and  general  principles,  139.  66,  Elimina- 
tion by  substitution,  141.  67,  Examples,  143.  68,  Elim- 
ination by  comparison,  144.  69,  Examples,  145.  70, 
Elimination  by  addition  and  subtraction,  146.  71,  Special 
expedients,  148.  72,  Examples,  149.  73,  Simultaneous 
equations  containing  three  unknown  numbers,  152.  74, 
Examples,  155.     75,  Literal  simultaneous  equations,  157. 

76,  Problems  producing  simultaneous  equations,  158. 

CHAPTER  Xn. 
Powers  and  Roots, --      163 

77,  Powers  of  monomials,  163.  78,  Square  of  a  binomial, 
168.  79,  Cube  of  a  binomial,  169.  80,  Square  of  a  poly- 
nomial, 171.  81,  Roots  of  monomials,  173.  82,  Examples, 
177.  83,  Square  root  of  polynomials,  179.  84,  Cube 
root  of  polynomials,  184. 

CHAPTER  Xin. 
Harder  Factors,  Multiples,  and  Fractions,       -        -        -      187 

85,  Factors  common  to  all  the  terms  of  an  expression,  187. 

86,  Formation  of  certain  products,  189.  87,  Expressions 
of  the  form  X'—a^,  191.  88,  Expressions  of  the  form 
x^-\-ax-\-l>,  193.  89,  Expressins  of  the  form  a^—b^^  195. 
90,  Expressions  of  the  form  a^-\-b^,  197.  91,  Expressions 
of  the  form  x^-\-a-x^-\-a^,  198.  92,  Expressions  of  the 
form  a^—b^y  200.  93,  Expressions  of  the  form  a"-\-b'\  208. 
94,  Miscellaneous  factors,  212.  95,  H.C.F.  of  expressions 
which  can  be  factored,  216.  96,  H.C.F.  of  expressions  not 
easily  factored,  217,  examples,  225.  97,  L.C.M.  of  expres- 
sions that  can  be  factored,  227.  98,  L.C.M.  of  expressions 
not  easily  factored,  229.     99,  Fractions  reduced  to  lowest 


viii  CONTENTS. 

terms,  231.  100,  Addition  of  fractions,  233.  lOl,  Sub- 
traction of  fractions,  234.  102,  Multiplication  of  fractions, 
236.  103,  Division  of  fractions,  238.  104,  Miscellaneous 
fractions,  240. 

CHAPTER  XIV. 

Quadratic  Equations, 249 

105,  Preliminary  topics,  249.  106,  Pure  quadratic  equa- 
tions, 252;  examples,  253.  107,  Affected  quadratics,  255; 
examples,  257.  108,  Problems  leading  to  quadratic  equa- 
tions,  261.     109,    Equations    solved   like   quadratics,  269. 

110,  Theory  of  quadratic  equations,  270. 

CHAPTER  XV. 
Simultaneous  Equations  Above  the  First  Degree,      -        -      277 

111,  One  equation  of  the  first  degree  and  one  of  the  second 
degree,  277;  examples,  279.  112,  Tv^^o  quadratic  equations, 
280;  examples,  282.  113,  Miscellaneous  equations,  283; 
examples,  286.     1 14,  Problems,  287. 

CHAPTER  XVI. 

Theory  of  Indices, 291 

115,  Meaning  of  fractional  exponents,  291,  116,  Examples, 
295.  117,  Properties  of  fractional  exponents,  296.  118, 
Examples,  299.  119,  Meaning  of  zero  and  negative  ex- 
ponents, 303.  120,  Examples,  305.  121,  Properties  of 
negative  exponents,  306.     122,  Examples,  310. 

CHAPTER  XVII. 

Surds,       - -314 

123*,  Definitions  and  general  principles,  314.  124,  To  re- 
move a  factor  from  beneath  the  radical  sign,  316.  125,  To 
introduce  the  coefficient  of  a  surd  under  the  radical  sign, 
317.  126,  To  integralize  the  expression  under  the  radical 
sign,  318.  127,  To  lower  or  raise  the  index  of  a  surd,  320. 
128,  To  reduce  a  surd  to  its  simplest  form,  320.  129, 
Addition  and  subtraction  of  surds,  321.  130,  Multiplica- 
tion and  division  of  surds,  323.  131,  Powers  and  roots  of 
sures,  326.  132,  Rationalization  of  expressions  containing 
quadratic  surds,  327.    133,  Rationalization  of  equations,  330. 


CONTENTS.  IX 

CHAPTER   XVIII. 

Ratio,  Proportion  and  Variation, 332 

134,  Ratio,  332;  problems,  337.  135,  Proportion,  339; 
problems,  344.  136,  Variation,  345;  examples  and  prob- 
lems, 348. 

CHAPTER   XIX. 

Progressions, 350 

137,  Arithmetical  progressions,  350.  138,  Examples  and 
problems,  354.  139,  Geometrical  progressions,  357.  140, 
Examples  and  problems,  360. 

CHAPTER   XX. 

Binomial  Theorem, 363 

141,  Laws  of  exponents  and  coefficients,  363.  142,  Ex- 
amples, 370. 


SCHOOL  ALGEBRA. 


CHAPTER  L 
FIRST  PRINCIPLES. 

EXERCISE   1. 

Illustrating  how  a  Letter  may  be  used  to  Represent  a  Number. 

1.  In  Algebra,  as  in  Arithmetic,  the  symbol  for  Plus 
is  -f-  and  the  symbol  for  Minus  is  — . 

1.  How  many  dozen  are  6  dozen  +  4  dozen  4-  2 
dozen  ? 

2.  How  many  score  are  6  score  +  4  score  +  2  score  ? 

3.  How  many  hundred  are  6  hundred  +  4  hundred 
+  2  hundred  ? 

4.  How  many  times  100  are  6  times  100  +  4  times 
100  +  2  times  100  ? 

5.  How  many  times  10  are  6  times  10  +  4  times  10 
+  2  times  10  ? 

6.  How  many  times  7  are  6  times  7  +  4  times  7  + 
2  times  7  ? 

7.  Six  times  a7ty  number  plus  four  times  the  same 
n^imbcr  plus  two  times  the  same  number  are  how  many 
times  that  number  f 

2,  In  Algebra  letters  are  often  used  to  represent  or  stand 
for  numbers. 


.2  FIRST    PRINCIPLES. 

3.  In  Algebra,  as  in  Arithmetic,  the  symbol  for 
times  is  x . 

In  any  statement  like  6X100+3X100—5X100,  /.  c,  in  any  state- 
ment where  addition,  or  subtraction,  and  multiplication  occur  to- 
gether, the  multiplications  must  always  be  performed  before  any 
addition  or  subtraction  takes  place.  Thus,  6X100-(-3  does  not  mean 
6X103,  but  means  600+3. 

8.  If  /  stands  for  10,  how  many  times  10  are  4x/+ 
7x/— cSx/? 

g.  If /stands  for  2,  how  many  times  2  are  4x/+7x/ 
— 8x/? 

ID.  If  t  stands  for  3,  how  many  times  3  are  4x/+ 
Tx/— 8x/? 

11.  If  5  stands  for  6,  how  many  times  6  are  7x^+ 
5X5— 3x^? 

12.  If  5  stands  for  7,  how  many  times  7  are  7X5+ 
5X5—3X5? 

13.  Seven  times  a  certam  number  plus  five  times  the 
safne  number  minus  three  times  the  same  number  are  how 
many  times  that  mimber  f 

14.  If  n  stands  for  a  certain  number,  how  ma^y  times 
that  number  are  7  X  ?^  +  5  X  ;z— 3  X  ;^  ? 

15.  In  question  14  can  n  stand  for  17?  for  100?  for 
25?  fori?  for  J?  for  li? 

In  qjiestion  z^,  71  ca7i  stand  for  Ki^Y  number  WHATEVER, 
provided  it  stands  for  the  same  niunber  throughoitt  question 
and  answer. 

16.  If  a  stands  for  a  certain  number,  8x«  +  4x<2— 
5  X  <2  are  how  many  times  that  number  ? 

17.  If  b  stands  for  a  certain  number,  8x<^+4x/^— 
hy.b  are  how  many  times  that  number ? 

18.  Could  some  other  letter  than  a,  b,  n,  5,  or  /  be 
used  to  represent  a  number  ? 


ALGEBRAIC    EXPRESSIONS.  3 

4.  The  preceding  questions  suggest  the  following 
principle  : 

A7iy  letter  may  be  icsed  to  represent  or  stand  for  any 
number,  provided  that  the  same  letter  represents  the  sa?ne 
number  throughout  the  saine  question  and  answer. 

6.  In  Algebra  it  is  usual  to  omit  the  sign  X  in  a 
product  like  7  X  w  and  write  merely  7«.  It  is  then  read 
*'  seven  n,''  instead  of  "seven  times  ^^,"  but  of  course  it 
always  mea^is  seven  times  ?^ ;  and  this  meaning  the  learner 
must  keep  in  mind.  Thus,  if  b  stands  for  31^,  then  7^ 
stands  for  7  times  Z\\. 

Evidently  when  the  sign  X  occurs  between  figtires  it 
cannot  be  omitted.  Thus  we  can  write  ^ib  for  7x<^, 
but  we  cannot  write  731|  for  7x31|-,  for  731|  has  a  dif- 
ferent meaning  already  given  to  it. 

6.  The  number  written  before  a  letter  to  show  how 
many  times  the  number  represented  by  the  letter  is  taken, 
is  called  the  Coefficient  *  of  the  letter.  Thus,  in  "b,  7 
is  called  the  coefficient  of  b. 

ig.  What  does  hb  mean  ?  How  much  is  this  if  b  equals 
10  ?     What  is  the  5  called  ? 

20.  What  does  10?/  mean  ?  How  much  is  this  if  n 
equals  8  ?     What  is  the  10  called  ? 

EXERCISE  2. 

Leading  to  the  Idea  of  an  Algebraic  Expression. 

1.  What  does  7a— 3^  +  5<2  equal,  \i  a  stands  for  7? 

2.  What  does  10i^4-4a— 9a  equal,  if  a  stands  for  6? 

*  A  more  general  definition  of  coefficient  is  given  in  Art.  14. 


4  FIRST  PRINCIPLES. 

3.  What  does  S5a—7a—Sa  equal,  if  a  stands  for  y\? 

4.  What  does  9<2  + 6a— 7a— 4a  +  2a  equal,  if  a  stands 
for  A? 

5.  What  does  25;^  +  |  equal,  if  7z  stands  for  |-? 

6.  What  does  2o?i—6n—^  equal,  if  n  stands  for  ^? 

7.  Anything,  whether  short  and  simple  or  long  and 
complicated,  which  is  or  may  be  considered  to  be  equal 
to  some  number,  is  called  an  Expression. 

The  number  to  which  the  expression  is  equal  is  called 
the  Value  of  the  Expression. 

The  number  which  a  letter  stands  for  is  called  the 
Value  of  the  Letter. 

7.  What  is  the  value  of  the  expression  10c-\-2c—5c— 
6c-hSc,  if  the  value  of  ^  is  4  ? 

8.  What  is  the  value  of  the  expression  12m—bm~Q>7}t 
-\-Sm,  if  the  value  of  m  is  12  ? 

9.  What  is  the  value  of  the  expression  7a— 4a— 3a  +  a, 
if  the  value  of  a  is  2  ? 

10.  What  is  the  value  of  the  expression  16^+24^— 
10^+25,  if  the  value  of  d  is  yV? 

EXERCISE  3. 

Leading  to  the  Notion  of  an  Algebraic  Equation  and  the  Dis- 
tinction BETWEEN  Known  and  Unknown  Numbers. 

8.  In  Algebra,  as  in  Arithmetic,  the  symbol  for 
Equals  is  =. 

1.  What  is  the  value  of  12a,  if  a= 2?  if  a=5?  ifa=7? 
ifa=|?  if  a=6i? 

2.  What  must  a  equal  if  the  value  of  12a  is  36  ?  if  the 
value  of  12a  is  48  ?  if  the  value  of  12a  is  72  ?  if  the  value 
of  12a  is  8  ?  if  the  value  of  12a  is  20  ? 


ALGEBRAIC    EXPRESSIONS.  5 

3.  If  ^=8,  what  does  Ix  equal?  If  7jtr=56,  what 
does  X  equal  ?     If  lx=4S),  what  does  x  equal  ? 

4.  If  ;«r=9,  what  does  \\x  equal?  If  ll;r=99,  what 
does  X  equal  ?     If  lljtr=121,  what  does  x  equal  ? 

5.  If  a=12,  what  does  3^-h5«  equal?  If  3^-f-5«=96, 
what  does  a  equal?     If  3«-f5a=4,  what  does  a  equal  ? 

6.  If  ;z=6,  what  does  6?i-^5n-\-7i  equal?  If  G7i-\-d?i 
+  «  =  72,  what  does  n  equal?  If  G?^  +  5;^^- 7^=108,  what 
does  ?i  equal  ? 

7.  If  7<^— 6<^  +  3^=20,  what  is  the  value  of  d? 

9.  The  statement  of  equality  which  exists  between 
two  expressions  is  called  an  Equation,  and  the  parts  on 
either  side  of  the  sign  =  are  called  the  Members  of  the 
equation.  The  expression  on  the  left-hand  side  of  the  sign 
=  is  called  the  Left  or  First  Member,  and  the  expres- 
sion on  the  right-hand  side  of  the  sign  =  is  called  the 
Right  or  Second  Member. 

8.  If  3>&4-2/l'  +  /t=120,  what  is  the  value  of  k? 

9.  If  4;r-f-5;f— 3;f=3,  what  is  the  value  of -r? 

10.  If  7jt:— 2jf-f-;f=6,  what  is  the  value  of;*;? 

11.  If2j'=2^,  what  is  the  value  of  jk? 

12.  If  3^+2^=12,  what  is  the  valve  of -^? 

13.  If  r=f,  what  is  the  value  of  12c-{-^—0c} 

14.  lfd=l^,  what  is  the  value  of  8^^+5^—10  ? 

15.  If/=1.2,  what  is  the  value  of  20/— 7/— 3/'? 

16.  If  lOOze^— 78ze/+15ze'=74,  what  is  the  value  of  rr? 

17.  If  17«—5?<— 4/^=66,  what  is  the  value  of  7^? 

18.  If  ^=5,  what  is  the  value  ot  25^—5^+5  ? 

19.  If  /i=100,  what  is  the  value  qf  20/i-6/i-3/i  +  100? 

20.  If  18z'-|-llz;4-2z/4-9z/=80,  what  is  the  value  of -j? 


6  FIRST    PRINCIPLES. 

10.  A  careful  inspection  of  the  questions  of  this  ex- 
ercise shows  that  when  there  is  only  one  letter  in  an 
expression,  two  cases  may  arise  :  first,  the  value  of  the 
letter  may  be  given  and  the  value  of  the  expression  re- 
quired ;  second,  the  value  of  the  expression  may  be  given 
and  the  value  of  the  letter  required. 

In  the  first  case  the  value  of  the  letter  is  known  or 
give?i,  and  in  the  second  case  the  value  of  the  letter  is 
unktiown  or  required. 

Thus  w^e  see  that  in  Algebra  there  are  two  kinds  of 
numbers,  called  respectively  Known  and  Unknown, 
either  of  which  may  be  represented  by  a  letter;  and  as  it 
is  possible  that  both  kinds  of  numbers  will  appear  in  the 
same  question,  it  is  customary  to  distinguish  between 
them  by  representing  the  known  numbers  by  the  first  and 
intermediate  letters  of  the  alphabet,  and  the  unknown 
numbers  by  the  last  letters  of  the  alphabet. 

EXERCISE  4. 

Symbolic  Expressions. 

1.  A  coat  and  hat  cost  |24  ;  the  hat  cost  $4.  What 
does  $24- $4  stand  for? 

2.  A  coat  and  hat  cost  $24  ;  the  hat  cost  %x.  What 
does  124 -Ix  stand  for  ? 

3.  A  coat  and  hat  cost  $24  ;  the  hat  cost  %x  and  the 
coat  5  times  as  much  as  the  hat.  What  does  %hx  stand 
for? 

What  does  $24-|5x  stand  for  ? 

What  does  %x-\-%bx,  or  %^x,  stand  for? 

4.  If  n  stands  for  a  certain  number,  what  exprcssicn 
will  stand  for  a  number  which  is  10  larger  ? 


PROBLEMS.  7 

5.  1{  a  stands  for  a  certain  number,  what  v/ill  stand 
for  a  number  which  is  25  smaller  ? 

6.  There  are  two  numbers ;  the  second  number  is  five 
more  than  twice  the  first  number.  If  71  represents  the  first 
number,  what  expression  will  represent  the  second  num- 
ber? 

7.  Of  two  numbers  the  second  is  12  less  than  5  times 
the  first.  If  n  stands  for  the  first  number,  what  will 
stand  for  the  second  number  ? 

8.  How  many  feet  in  n  yards  ? 

9.  How  manj^  feet  in  ?i  yards  plus  5  A-ards  ? 

10.  How  many  feet  in  71  yards  plus  5  feet  ? 

11.  A  room  is  a  yards  wide  and  twice  as  long  as  it  is 
wide.  How  many  yards  long  is  the  room  ?  How  many 
feet  long  is  the  room  ? 

12.  In  jr  years  a  man  will  be  36  years  old.  What  is 
his  present  age  ? 

13.  A  man  is  now  40  years  old.  How  old  will  he  be 
a  years  from  now  ?  How  old  will  he  be  '2a  years  from  now  ? 
How  old  was  he  Sd  years  ago  ? 

EXERCISE  5. 

Problems. 

I.  A  coat  and  hat  cost  $24.  The  coat  cost  5  times  as 
much  as  the  hat.     What  was  the  cost  of  each  ? 

SOLUTION  BY  ARITHMETIC. 

Five  times  ^/le  cost  of  the  hat  =  the  cost  of  the  coat. 
Once  the  cost  of  the  hat  =  the  cost  of  the  hat. 
Therefore,  six  times  the  cost  of  the  hat  —  the  cost  of  coat  and  hat. 

But  the  cost  of  the  coat  and  hat  is  $24. 
Hence,         six  times  the  cost  of  the  hat  =  $24. 
Therefore,  the  cost  of  the  hat  =  ^  of  $24,  or  $1 

Consequently  the  cost  of  the  coat  was  $20. 


8  FIRST    PRINCIPLES. 

SOLUTION   BY  ALGEBRA. 

The  cost  of  the  hat  is  a  certain  number  of  dollars,  and  that  number, 
whatever  it  is,  we  may  represent  by  a  letter.  For  example,  we  may 
say, 

Let  X  =  number  of  dollars  the  hat  cost. 

Then  5x  =  number  of  dollars  the  coat  cost. 

Hence,  5x-^x,  or  Qx,  =  number  of  dollars  that  both  hat  and  coat  cost. 
But  24  =  number  of  dollars  that  both  hat  and  coat  cost. 

Therefore  6^  =  24. 

Then  x  =  4, 

and  5x  =z  20. 

Therefore  the  hat  cost  $4,  and  the  coat  cost  $20. 

The  student  should  compare  very  carefully  the  two  solutions  of  this 
problem  above  given.  It  will  be  noticed  that  the  arithmetic  and 
algebraic  solutions  are  not  so  different  from  each  other  as  would 
appear  at  first  sight.  In  fact,  to  change  the  first  solution  to  the  sec- 
ond nothing  need  be  done  except  to  replace  the  words  "  i/ie  cost  of  the 
hat "  by  $x.  In  the  arithmetic  solution  the  phrase  "  the  cost  of  the  hat'' 
stands  for  a  certain  unknown  number  of  dollars,  which  number  of 
dollars  is  represented  by  a  single  letter,  x,  in  the  algebraic  solution. 

2.  The  sum  of  two  numbers  is  72  and  one  number  is 
twice  as  large  as  the  other.     What  are  the  numbers  ? 

Let  jr=:the  smaller  number. 

Then  2x=the  larger  number. 

Therefore  'lx-^rx—Tl, 
i.  e.  3x=72. 

Hence  .^•— 24,  the  smaller  number, 

and  2.*"=48,  the  larger  number. 

3.  Divide  $65  between  A  and  B  so  that  B  shall  receive 
4  times  as  much  as  A. 

4.  A  rectangle  is  3  times  as  long  as  it  is  broad,  and 
the  distance  around  it  is  64  feet.  Find  the  length  and 
breadth  of  the  rectangle. 

5.  A  piece  of  timber  18  feet  long  must  be  cut  so  as  to 
give  one  piece  2  feet  long  and  two  other  pieces,  one  of 
which  must  be  3  times  the  length  of  the  other.  Find 
how  long  each  one  of  these  pieces  will  be. 


PROBLEMS.  9 

6.  A  father  said  to  his  son,  "  Neiit  year  the  sum  of  our 
ages  will  be  70  years,  and  I  will  be  4  times  as  old  as  you 
will  be."    What  is  the  present  age  of  each? 

7.  A  man  has  $48,  consisting  of  an  equal  number  of 
bank  notes  of  the  denominations  of  $1,  $2,  and  85.  What 
number  has  he  of  each  ? 

Let  j:=the  number  he  has  of  each.  Then  in  $1  bills  he  has  $x, 
in  $2  bills  he  has  $2x,  and  in  $5  bills  he  has  $5^-. 

8.  John,  Henry,  and  Mary  paid  13.60  for  their  books. 
Henry's  cost  twice  as  much  as  John's,  and  Mary's  cost 
three  times  as  much  as  John's.  Find  the  cost  of  each 
scholar's  books. 

9.  If  you  add  together  3  times  and  5  times  and  7  times 
a  certain  number  you  will  obtain  315.  What  is  the 
number  ? 

10.  Three  persons  subscribed  150000  to  build  a  hospital. 
The  first  two  subscribed  equal  amounts,  but  the  third  party 
subscribed  2  times  as  much  as  either  of  the  others.  How 
much  did  each  person  subscribe  ? 

11.  A  man  bought  10  turkeys,  10  chickens,  and  10 
ducks,  paying  §10  for  all.  A  chicken  cost  the  same  as  a 
duck,  and  a  turkey  cost  3  times  as  much  as  a  dnck.  What 
was  the  cost  of  each  ? 

12.  A,  B,  and  C  form  a  partnership  to  do  business.  A 
furnishes  4  times  as  much  capital  as  C,  and  B  furnishes  3 
times  as  much  as  C.  Altogether  the  three  men  put  in 
$24000.    Wliat  amount  is  furnished  by  each  ? 

13.  A  man  and  three  boys  did  a  piece  of  work  for  $16. 
How  should  this  money  be  divided  among  them,  if  we 
suppose  that  the  man  did  twice  as  much  work  as  each 
boy? 


lO  FIRST    PRINCIPLES. 

14.  A  man  has  a  farm  of  240  acres,  of  which  there  is 
twice  as  much  marsh  land  as  wood  land,  but  the  rest  of 
the  farm  is  3  times  as  large  as  the  marsh  land  and  wood 
land  taken  together.  Find  the  number  of  acres  each  of 
marsh  land  and  wcod  land  in  the  farm. 

15.  Divide  $1100  among  A,  B,  and  C  so  that  A  may 
have  twice  as  much  as  B,  and  B  three  times  as  much 
as  C. 

16.  A  man  raised  1730  bushels  of  grain,  of  which  there 
was  3  times  as  much  oats  as  wheat,  and  2  times  as  much 
corn  as  oats.  Find  the  amount  of  each  kind  that  he 
raised. 

17.  On  a  certain  day  a  storekeeper  took  in  $3G0,  of 
which  there  was  4  times  as  much  in  bank  notes  as  in 
coin,  and  5  times  as  much  in  silver  as  in  gold.  Find  the 
amount  of  each  kind  of  money  that  he  received. 

18.  Of  three  brothers  the  middle  one  is  4  times  and 
the  oldest  one  5  times  as  old  as  the  youngest.  The  sum 
of  the  ages  of  the  two  oldest  is  36  year?.  Find  the  age 
of  each  of  the  brothers. 

ig.  If  each  year  I  should  double  the  money  that  I  had 
at  the  beginning  of  that  year,  in  five  years  from  now  I 
would  have  $63000.     How  much  money  have  I  now  ? 

20.  There  are  three  numbers,  the  sum  of  the  last  two 
of  which  is  63.  The  second  is  five  times  the  first,  and 
the  third  equals  the  difference  between  the  Jfirst  and 
second.     What  are  the  numbers  ? 

EXERCISE  C. 

Expressions  in  which  Several  Letters  ars  used  to  stand 
FOR  Numbers. 

11.  We  have  already  learned  that  any  letter  may  be 
used  to  represent  any  number,  but  it  often  happens  that 


EXPRESSIONS   WITH    SEVERAL   LETTERS.  II 

different  numbers  occur  in  the  same  question.  In  this 
case  different  letters  may  be  used  to  represent  these  num- 
bers, but  here,  as  before,  any  07ie  letter  must  represent 
the  sa7ne  number  throughout  question  .an^  answer. 

P^ind  the  value  of  the  following  expressions,  if  «=5, 
^=4,  ^=8,  ^=2,  ^=1: 

1.  'da-\-2b-\-Ze—2e,  4.  ha—b—c—Zd—Q,e. 

2.  Ae+?^a—U+bc.  5.   I0e—2e+^b—a^d. 

3.  2a-\-'ib-e-c+l.  6.  20a^-6^-2«. 

12.  Just  as  Q>xb  is  v/ritten  6^,  so  if  we  wish  to  ex- 
press the  product  of  two  numbers  represented  by  a  and  b 
respectively,  we  would  write  it  merely  ab,  instead  oi axb. 
Also  abe  means  just  the  same  as  axbxc.  This  custom 
of  omitting  the  sign  X  between  a  figure  and  a  letter,  or 
between  two  letters,  is  universal  in  Algebra. 

Find  the  value  of  each  of  the  following  six  expressions, 
if  fl;=2,  ^=4,  r=6,  d=^^,  e=d  : 

7.  7ae-\-Sbe+de.  10.  4abcd-^Sbcd—cde-\-5d—S-^. 

8.  Sabd—cd—ab+2.        11.   bce+idab— bade +10— be. 

9.  2bed-\-See—a—bbde.    12.  abede+10  +  bed+9—dee-\-8. 
Find  the  value  of  each  of  the  following  eight  expres- 

rions,  if  a==6,  b=^,  e=4,  n=\,  s=l,  t=^  : 

13.  10at—2ens-\-6^—at.   17,  aa-\-ab+ast-\-ee. 

14.  ab-\-be?it—4:S-\-20?i.     18.   eee-\-aa-^ss—4ae. 

15.  3et—7eii-\-2e7i—abt.    19.   aa—2a-\-eee—Se-\-ssss—is. 

16.  Ses— 2 be— et-\- 12^.      20.   aaee— 500— 7teee—6ab. 

21.  What  is  the  value  of  <^<^^,  if  <^=  3?  if  ^=5?  if/^=i? 
if^=i?if^=|? 

22.  If  <2=2,  what  is  the  value  of  aa  ?  of  aaa  ?  of  aaaa  ? 
of  aaaaa ?  of  aaaaaa ? 


12  FIRST    PRINCIPLES. 

13.  When  a  product  consists  of  the  same  number  re- 
peated any  number  of  times,  the  product  is  called  a 
Power  of  that  number,  and  is  usually  written  in  a 
simplified  form.     Thus  : 

aa  is  written  a^ ,  read  a  square  or  second  power  of  a\ 

aaa  is  written  a"^ ,  read  a  cube  or  third  power  of  a; 

aaaa  is  written  a*,  read  a  fourth  ox  fourth  power  of  a\ 

and  so  on. 

The  small  figure  written  above  and  to  the  right  of  a 
number  is  called  the  Exponent  or  Index  of  the  power  ; 
it  shows  how  many  times  the  number  occurs  in  the 
power. 

According  to  this  notation,  a"^  would  mean  that  a  is  used  once  as  a 
factor,  but  when  a  number  is  used  only  once  as  a  factor  it  is  cus- 
tomary to  omit  the  exponent  altogether  and  write  simply  a  instead 
of  «!. 

23.  Write  3  times  x  square  ;  5  times  y  cube  ;  5  times 
the  fourth  power  oi  b  \  7  times  the  fourth  power  of  (^  ;  9 
times  the  fourth  power  oi  b\  a  times  the  fourth  power  of  ^. 

24.  How  would  you  abbreviate  bbxxx  ? 
Just  as  ay^h  is  written  ab,  so  b*Y.x^  is  written  b'^x*. 

25.  Write  in  the  abbreviated  form  expressions  17  to 
20  inclusive. 

26.  Find  the  value  of  ^a'^b'^-x^,  if  «=4,  ^=3,  and 
x=^\. 

Find  the  value  of  each  of  the  following  five  expressions, 
where  a=2,  <^=3,  ^=1,  ^=6  : 

27.  «2  4.32^^2_|.^2^  29.  a^J^b^—c^. 

28.  2^2+3^^-4^^  30.   '^abc—b'^c—iSc^. 

31.       a3^3«2^  +  3«^2_|_^3^ 

14.  When  several  numbers  are  multiplied  together 
the  result  is  called  the  Product,  and  each  of  the  numbers 


EXPRESSIONS   WITH    SEVERAL   LETTERS.  I  ^. 

or  the  product  of  any  number  of  them  is  called  a  Factor 
of  the  product.  Thus,  if  5,  a,  b,  b  are  multiplied  to- 
gether the  product  is  oab- ,  and  the  factors  of  the  product 
are  5,  a,  b,  b"^,  5a,  5b,  ab,  5ab,  5b'^,  ab'^. 

Any  factor  of  an  expression  is  called  the  Coefficient: 
or  Co-factor  of  the  product  of  the  remaining  factors.. 
Thus,  in  the  expression  a'^bc,  a"^  is  the  coefficient  or  co- 
factor  of  be,  a'^b  is  the  coefficient  of  r,  ab  is  the  coefficient 
oi  ac,  a'^c  is  the  coefficient  of  b,  and  a  is  the  coefficient  of 
abc.  Similarly,  be,  e,  ae,  b,  and  abe  are  the  coefficients  of 
a'^,  a'^b,  ab,  a'^e,  and  a,  respectively. 

When  a  product  is  made  up  partly  of  numbers  repre- 
sented by  figures  and  partly  of  numbers  represented  by  let- 
ters, as  2  X  ^ab'^,  it  is  sometimes  convenient  to  distinguish, 
between  these  two  kinds  of  factors.  In  this  case  we  call 
the  numbers  represented  by  figures  nitmerieal  factors  and 
those  represented  by  letters  literal  factors. 

Of  course  the  product  of  all  the  numerical  factors  is  the 
coefficient  of  the  product  of  all  the  literal  factors,  and  the 
former  is  often  referred  to  as  the  Numerical  Coefficient. 

32.  In  abx"^  what  is  the  coefficient  of  x'^l  of  bx'^1  of 
ab"^.   of  abx?  of  a  ? 

33.  In  12abn  what  is  the  numerical  coefficient?  of 
what  is  it  the  coefficient  ?  What  is  the  coefficient  of  bn  ? 
of2b?i}  o{Aab?i?  of  3? 

34.  Write  five  different  factors  of  15edx^,  and  tell 
of  what  each  factor  named  is  the  coefficient. 

35.  Write  five  different  factors  of  ^a^?tj'^,  and  tell  of 
what  each  factor  named  is  the  coefficient. 

15.  In  Algebra,  as  in  Arithmetic,  the  symbol  for 
Divided  by  is  -r-. 


14  FIRST   PRINCIPLES. 

More  often,  however,  division  is  represented  by  means 
of  a  fraction  where- the  dividend  is  written  above  the  line 
and  the  divisor  below. 

Thus,  a-^d  is  written  7-,  and  v/hcn  written  in  this  form 
0 

it  is  often  read  "a  over  3." 

Write  the  following  in  the  usual  algebraic  notation,  as 
explained  in  Arts.  10,  13,  and  14. 

36.  A  known  number  divided  by  G. 

37.  A  known  number  divided  by  anotjier  knov/n 
number. 

38.  An  unknown  number  divided  by  three  times  a 
known  number. 

39.  Four  times  the  square  of  a  knoAvn  number  divided 
by  5  times  the  cube  of  an  unknown  number. 

40.  Twice  the  square  of  a  known  number  times  the 
cube  of  an  unknown  number  over  the  product  of  two 
unknown  numbers. 

Find  the  value  of  each  of  the  following  expressions, 
if  ^=2,  3=5,  m=S,  s=-l  ^=|: 

8ab  .  403  ^    ah?i  ,    3       ^ 


ow 


3m  b        has 


^2     in  9^-/  ,  b'^s^in   ,  3 

42. V-f.  47.   -^ — V — r f-T 

171       b  2            5          4 

VI       b  _    1    ^         1    ,  ,  tm 

43.  —„-  +  —.  48.  ^a■'m  —  ^ab^ 

a^      in  L             6           a 

a'^       b  ^2       3       3^2^ 

a''       b  a        t       T) 

^      b       a^  s"      in      12 


CHAPTER  II. 

UNION  OF  TERMS  AND  REMOVAL  OF 
PARENTHESES. 

EXERCISE  7. 

Union  of  Similar  Terms. 

16.  When  an  expression  is  broken  in  parts  just  before 
each  of  the  signs  +  and  — ,  each  part  thus  formed  is 
called  a  Term  of  the  expression.  Thus,  in  the  expres- 
sion Sad-^Ac"^ ~2e—lo,  the  terms  are 

Sad,  -i-ic\  -2e,  -15. 
We  speak  of  the  terms  of  an  expression  in  a  manner  somewhat 
analagous  to  the  way  in  which  we  speak  of  the  syllables  of  a  word. 
A  syllable  may  not  convey  an  intelligible  idea  when  taken  by  itself, 
but  when  joined  with  other  syllables  to  make  up  a  word,  the  whole 
word  does  convey  a  definite  idea.  So  a  /tv;//  taken  by  itself  may  not, 
at  this  stage,  express  any  idea,  but  the  whole  expression  does  convey 
an  idea.  See  Art.  7.  Indeed,  each  term  would,  even  now,  convey  a 
definite  idea  were  it  not  for  the  sign  -|-  or  —  which  always  goes  with 
each  term  after  the  first. 

17.  The  terms  of  an  expression  which  have  the  sign 
-|-  or  no  sign  at  all  are  called  Additive  Terms,  and  those 
terms  which  have  the  sign  —  are  called  Subtractive 
Terms.  Thus,  in  the  expression  n-i-2d—4a  +  5a—7d, 
the  terms  ;^,  -\-2d,  and  -{-oa  are  additive  terms,  and  —4a 
and  —7d  are  subtractive  terms. 

In  comparing  several  additive  terms  they  are  spoken 
of  as  terms  of  the  same  s(^n,  and  several  subtractive  terms 
are  also  spoken  of  as  terms  of  the  same  sign  ;  but  when 


1 6  REMOVAL   OF    PARENTHESES. 

additive  and  subtractive  terms  are  spoken  of  together, 
they  are  said  to  be  terms  of  imlike  or  opposite  signs. 

Notice  that  when  we  speak  of  the  signs,  without  any  further  quali- 
fication, that  it  is  only  thejirst  tivo  of  the  four  fundamental  signs  of 
algebra,  -\-,  —,  X,  and  -f-,  that  we  have  reference  to. 

18.  Terms  whose  literal  parts  are  identical  and  whose 
signs  and  numerical  coefficients  may  or  may  not  differ  are 
called  Similar  Terms.  Thus,  in  the  expression  'i)a-d— 
?>ad^—a~d-\-4.ad'^,  the  terms  da'^d  and  —a'^b  are  similar; 
also  the  terms  —Zab'^  and  ■\-\ab''-'  are  similar;  but  ^a'^b  and 
+  4a^^  are  7iot  similar. 

1.  Is  the  expression  12  +  2—8  +  5—4 
the  same  as                                 12  +  2  +  5—8  —  4  ? 

Is  the  last  equal  to  19  —  8—4  ? 

Is  this  the  same  as  19  —  12  ? 

2.  Is  the  expression  2  +  4—5  +  20—14  +  11 
the  same  as  2  +  4  +  20  +  11-5-14? 

Is  the  last  equal  to  37  —  5  —  14? 

Is  this  the  same  as  37  — 19  ? 

3.  Is  the  expression  8^— 2a— 4«  +  5«— 2« 
the  same  as  8^  +  5^-2^— 4a— 2a? 

Is  the  last  equal  to  13a— 2a— 4a— 2a  ? 

Is  this  the  same  as  13a— 8a  ? 

4.  Is  the  expression  ;^  +  5a^^  —  2a(^^  +  ISaZ)^  —  7  —  12a<^2 
the  same  as  n^hab'' ^Ihab"- -lab'' -\1ab''^ -1 1 

Is  the  last  equal  to  7i-\-%)ab''- -lab'' —  Xlab'' -1 1 
Is  this  the  same  as  ;^  +  20a^2  — 14a<^-  — 7  ? 

5.  In  a7iy  expression  whatever^  will  the  value  of  the  ex- 
pression be  changed,  if  all  the  additive  terms  are  written 
first  and  the  subtractive  terms  following  ? 

6.  In  any  expression,  can  any  number  of  similar  addi-  . 
tive  terms  be  replaced  by  a  single  additive  term  ? 


UNION   OF    SIMILAR   TERMS.  1 7 

7.  In  any  expression,  can  any  number  of  similar  sub- 
tractive  terms  be  replaced  by  a  single  subtractive  term  ? 

8.  A  man  had  n  dollars.  He  gained  23  dollars  and 
then  lost  9  dollars.     How  much  did  he  then  have  ? 

How  much  more  is  /2-f  23— 9  than  71  ? 

Write  an  expression  of  two  terms  which  shall  be  equal 
to«  +  23-9. 

The  two  terms  +23—9  can  be  replaced  by  what  single 
term? 

9.  A  man  had  n  dollars.  He  gained  9  dollars  and  then 
lost  23  dollars.     How  much  did  he  then  have  ? 

How  much  less  is  w-j-9— 23  than  n  ? 

Write  an  expression  of  two  terms  which  shall  be  equal 
to  w  +  9-23. 

The  two  terms  -|-9— 23  can  be  replaced  by  w^hat  single 
term  ? 

10.  How  much  more  is  « +  52-21  than  a? 

Write  an  expression  of  two  terms  which  shall  be  equal 
to«  +  52-21. 

The  two  terms  +52—21  can  be  replaced  by  what  single 
term  ? 

11.  How  much  less  is  fl;  +  21— 52  than  «? 

Write  an  expression  of  two  terms  which  shall  be  equal 
to«  +  21-52. 

The  two  terms  +21—52  can  be  replaced  by  what  single 
term  ? 

12.  How  much  more  is  100+17^-5^  than  100? 
Write  an  expression  of  two  terms  which  shall  be  equal 

to  100  +  17«-5«. 

The  two  terms  ■\-VJa—ha  can  be  replaced  by  what  sin- 
gle term  ? 


1 8  REMOVAL    OF    PARENTUESES. 

13.  How  much  less  is  100  +  5a— 17^  than  100? 
Write  an  expression  of  two  terms  which  shall  be  equal 

to  100  +  5^— 17a. 

The  two  terms  +  5a— 17a  can  be  replaced  by  what  sin- 
gle term  ? 

14.  How  much  more  is  n-\-2da—lla  than  ;?  ? 

Write  an  expression  of  two  terms  which  shall  be  equal 
to  ;z  +  29a  — 11a. 

The  two  terms  +  20a  — 11a  can  be  replaced  by  what  sin- 
gle term  ? 

15.  How  much  less  is  ;z  +  lla— 29a  than  n? 

Write  an  expresGion  of  two  terms  v/hich  shall  be  equal 
to  ;^  +  lla— 29a. 

The  two  terms  +  11a— 29a  can  be  replaced  by  what  sin- 
gle term  ? 

16.  In  any  expression  whatever,  can  two  similar  terms, 
one  additive  and  one  subtractive,  be  replaced  by  a  single 
term? 

What  kind  of  a  term  is  this,  if  the  numerical  coefficient 
of  the  additive  term  is  the  greater  ? 

What  kind  of  a  term,  if  the  numerical  coefficient  of  the 
subtractive  term  is  the  greater  ? 

19.  The  above  questions  suggest  the  following  prin- 
ciples : 

/.  T/ie  order  of  terms  in  an  expression  is  not  fixed,  the 
value  of  an  expression  being  U7ichanged  if  all  the  additive 
terms  be  written  first,  followed  by  the  subtractive  teims. 

II.  Any  number  of  similar  additive  terms  in  an  expres- 
sion may  be  replaced  by  a  single  additive  teini,  and  any 
number  of  similar  subtractive  terms  may  be  replaced  by  a 
single  subtractive  term. 


EXAMPLES.  19 

///.    Two  similar  terms,  one  additive  and  one  snbtractive, 
may  be  replaced  by  a  single  term  in  which  the  munerical 
coefficient  is  the  difference  of  the  numcidcal  coefficients  of  the 
two  ter77is,  and  the  sign  is  -\-  or  —  the  same  as  that  07ie  of 
the  displaced  terms  havi^ig  the  gr'eater  numerical  coefficient. 

EXERCISE  8. 

Examples. 

In  each  of  the  following  expressions  combine  the  similar 
additive  terms  into  a  single  additive  term,  and  the  similar 
snbtractive  terms  into  a  single  snbtractive  term.  After- 
wards combine  these  tw^o  terms  into  a  single  term. 

1.  7^— 3a— 4«  +  6«.  4.   \^^-\-Zab—1ab-\-hab—Zab. 

2.  1U-— 5;»:-f8;t:— 6;i;.        5.  n-\-12y—lSy-{-2y—6y. 

3.  50-6^+2^-20^+8^^.  6.  8^-3^+7^-9^-4^+8^. 

7.  a  +  20xy—n0xy-o0xy-\-10xy. 

8.  a  +  9a2-2«2_^5a2  +  8a2_7«2_9^.2^ 

9.  37  —  10m-\-nm—15m—Sm. 

10.  r>m—4:7n-^S?n—12m—77n-\-10m. 

11.  10x^-7x^-j-Sx^  +  Sx^-12x^-x\ 

12.  125  +  8j2_8_>'2^-6y2_20>/2_4r2^ 

13.  10  +  5a2^2_^2^2_7^2^2^5^2^2_^2^2^ 

14.  n-\-20x'^y—6x''^y+x'^y—nx'^y—10xy. 

15.  100+13j;i;2-23jjr2+33j/jt:2-43j/Ji'2. 

16.  Ga— 18^2^+4<^V-^V+10^V. 

1 7 .  50a bed—  11a bed—  Ala bcd-\-  Sa bed. 

18.  ^x—^^x-\-x-\-fX. 

^X  -  {  -r+X+l  X  =  1  ^+|-;c+X  -  J  T, 

=^-^+«^+-6-^-ii^, 
=-V--^-|-^.  etc. 

19.  G  +  -Ja+|a-ia.  21.   5b+lb-ib-{-2b-U+Vj. 

20.  1— y^a  +  |a  +  ia— a.    22.   10a—Sb+ib+^b—b—\b. 


20  REMOVAL   OF    PARENTHESES. 

Shorten  the  following  expressions  as  much  as  possible 
by  a  careful  grouping  and  uniting  of  similar  terms  : 

23.  99997  +  83752  +  3. 

24.  9999  +  9998  +  9996  +  9095+4+2+1+5. 

25.  17  +  18  +  19  +  20+21  +  22  +  23. 

26.  Sa  +  7d-h5a.  23.  24/ +  30^+17^+35/. 

27.  x-i-oy+Sx.  29.   Ufi-\-27ad-\-8n-^lSad-^l(j?i. 
30.   Sa-h^d''  +Sdc-i-Gdc-\-17a  +  lU'- -i-dd^  +2a+dc. 

31.  5^  +  3^—3^.  36.  7a-j-6d—Aa. 

32.  9^  +  3«— 9^.*  37.  9a--bd-\-a. 

33.  4a— .r+A-.  38.  20— 7.r+9. 

34.  84589  +  8783-4589.  39-  57-7a-9. 

35.  28654  +  9999-18054.  40.  30+6r+5. 

41.  18«-16^^+10a+14^r-16^-2<^.- 

42.  lQa''—Sad-Sad-6a--hl2ad. 

43.  2xy—^2-\-\Qxy—82-\-1^2—12xy, 

44.  8w— 10A'  +  6;z— 4/5— 2;z  +  2w— 8/5— 6w. 

45.  5«  +  8/iV-7^-2«-9<^V+2^-2a  +  2^V+6^. 

46.  lOw  +  n— 5jr— 12— 4w— 3;t-+l  +  9x— 5;;/. 

47.  9a-7^+3^-8a  +  7^-3r-5/^-8r. 

48.  ^a^b^c-7a^b''c''-^10a^b''c''-ha^b^c, 

49.  llxy+2ab—Axy—2bab-\-ab-]-10, 

50.  25-25jc+2rj^'+13-30j^/+20ji;-8. 

51.  ;z2_2  +  ^2^2-16r2+9;i24.5_|.^2^ 

52.  lSx—5y+8z—5x+9y—llz—'6x—6y+2, 

53.  54w— 62/Z  +  18X— 62w— 6;r+42«  +  10w+18w— 14j»; 

54.  10;;z  +  ll-5«2_l2  +  4w-6a2  +  l  +  18a2_i0;;i. 

55.  ^-2/^^+18^^— 14^^-21<r^-3^^+5<:^. 

56.  9a4;r2  — 54a-5.;»;+27iJ^^—7a*;i;2  +  13^.^;»;— aS:r+3a«. 


REMOVAL  OF  PARENTHESES.  21 

EXERCISE  9. 

Parentheses  and  their  Removal. 

20.  If  we  wish  to  indicate  that  the  sum  of  the  num- 
bers 2  and  3  is  to  be  subtracted  from  12,  we  write  it  like 
this, 

12-(2  +  3), 
using  ^parenthesis  to  enclose  the  expression  2  +  3. 

The  use  of  the  parenthesis  in  Algebra  is  quite  frequent; 
and  it  always  serves  to  show  that  the  expression  which 
it  embraces,  whether  simple  or  complicated,  is  to  be  looked 
upon  as  a  single  number  just  as  though  it  were  represented 
by  a  single  symbol. 

Thus,  (3  +  2)  X  (7—5),  or  (3  +  2)(7— 5),  means  the  same 
as  5x2  ;  (3  +  2)^  means  the  same  as  5^,  that  is,  25  ;  also 
(3a) '^  means  the  square  of  o  times  a,  but  3a ^  means  3  tiynes 
the  square  of  a.  The  student  will  readily  notice  the  dif- 
ference between  (3a)  ^  and  3a  2. 

Expressions  containing  parentheses  should  be  read  in  a  manner  so 
as  to  show  what  numbers  are  enclosed  in  parentheses.  Thus, 
(3+2)(7  — 5)  may  be  read,  "parenthesis  3+2  multiplied  by  parenthesis 
7  —  5,"  but  should  not  be  read,  "3  plus  2  times  7  minus  5,"  for  this 
last  would  be  the  way  to  read  the  expression  3+2X7—5. 

1.  Write  16  diminished  by  the  sum  of  the  numbers  2, 
3,  and  4. 

2.  Write  a  diminished  by  the  sum  of  the  numbers  b, 
c,  and  d. 

3.  Write  3  times  a  minus  the  sum  of  the  numbers  2 
times  b,  3  times  r,  and  4  times  d. 

4.  Write  a  plus  the  square  of  the  sum  of  the  numbers 
b  and  e. 

Find  the  value  of  each  of  the  following  expressions, 
5  to  11,  where  ;;^  =  G,  /=5,  />=4,  ^=3,  and  r=2  : 


22  REMOVAL    OF    PARENTHESES. 

5.  (w-Ox(/  +  ^)-r.  7.  w  +  (/-/)-(^+r). 

6.  ^{rii^t^p)^{q-r).  8.    {^m-t)-{p^-q-\-r), 

9.   w  +  2(/-/')-(^-r). 

10.  (;;22  — /'2)_(^  +  2^)  +  r2. 

11.  2(;;^2_/2) +  1(^4.2^) -f;.2. 

Find  the  value   of  the   following  three  expressions, 
where  ^=1,  ^=2,  <:=3  : 

12.   (2^  +  ^)2+4(2^-2^)3.     13.  3(a  +  3)2-(^  +  r)2. 
14.   Qi{a-\-U)c—{b-\-c)abc. 

EXERCISE  10. 

Removal  of  Parentheseg. 

GENERAL  FORM  a-\-{b-\-c). 

1.  How  much  greater  is  10 +  (7 +  3)  than  10  +  7  ? 

2.  How  much  greater  is  12  +  (7  +  3)  than  12  +  7? 

3.  How  much  greater  is  25 +  (7 +  3)  than  25  +  7  ? 

4.  How  much  greater  is  «  +  (7  +  3)  than  a-j-7  ? 

5.  How  much  greater  is  «  +  (8  +  3)  than  a-j-8? 

6.  How  much  greater  is  «  +  (<5+3)  than  a-\-b} 

7.  How  much  greater  is  ^  +  (^  +  5)  than  a-j-d? 

8.  How  much  greater  is  a-\-(d-^c)  than  a-^-d? 

9.  How  much  must  be  added  to  «  +  ^  to  equal  «  +  (^-  /)? 

10.  Write  an  expression  equal  to  a  +  (d-\-c)   without 
using  a  parenthesis. 

GENERAL    FORM    a-^[d—c). 

11.  How  much  less  is  10  +  (7-3)  than  10  +  7  ? 

12.  How  much  less  is  12+ (7— 3)  than  12  +  7  ? 

13.  How  much  less  is  25  + (7—3)  than  25  +  7  ? 

14.  How  much  less  is  «  +  (7  — 3)  than  a +  7? 

15.  How  much  less  is  «  + (8— 3)  than  «  +  8? 


REMOVAL   OV    PARENTHESES.  23 

16.  How  much  less  is  a  +  (/^— 3)  than  a-{-d?    Why? 

17.  How  much  less  is  « -I- (<^— 5)  than  <2 4-^?    Why? 

18.  How  much  less  is  a-^(d—c)  than  a-\-d? 

19.  How  much  must  be  subtracted  from  a-\-l?  to  equal 

20.  Write  an  expression  equal  to  a-{-(d—c)  without 
using  a  parenthesis. 

GENERAL    FORM    a  —  [/>-\-c). 

21.  How  much  less  is  10-(7  +  3)  than  10—7? 

22.  How  much  less  is  12  — (7  +  3)  than  12—7? 

23.  How  much  less  is  25-(7  +  3)  than  25-7  ? 

24.  How  much  less  is  «— (7-f  3)  than  a— 7  ?     Why? 

25.  How  much  less  is  a— (8-;-3)  than  a—S?  Why? 
25.  How  much  less  is  «—(^ '1-3)  than  «—^?  Why? 
27.   Plow  much  must  be  subtracted  from  a— I?  to  equal 

a-(^  +  r)? 

23.   How  much  less  is  a—(d-\-iJ)  than  a—d"? 

29.  How  much  must  be  subtracted  from  a  —  d  to  equal 

30.  Write  an  expression  equal  to  «— ((^+5)  without 
using  a  parenthesis. 

31.  How  much  lees  is  a—(id-\-c)  than  a—d?     Why? 

32.  How  much  must  be  subtracted  from  a  —  d  to  equal 
a-(d+c)? 

33.  Write  an  expression  equal  to  ^— (/>  +  <:)  without 
using  a  parenthesis. 

34.  The  additive  terms  d  and  -i-c  within  the  parenthesis 
occur  as  what  kind  of  terms  when  the  expression  is  written 
without  a  parenthesis  ? 


24  '    REMOVAL    OF    PARENTHESES. 

GENERAL    FORM    a  —  [b  —  c). 

35.  How  much  greater  is  10— (7— 3)  than  10—7? 

36.  How  much  greater  is  12  — (7— 3)  than  12—7? 

37.  How  much  greater  is  25— (7— 3)  than  25—7? 

38.  How  much  greater  is  a—(J  —  Z)  than  a— 7  ? 

39.  How  much  greater  is  «— (8— 3)  than  «— 8  ? 

40.  How  much  greater  is  a  —  (^b—Z')  than  a—b} 

41.  Hov/  much  must  be  added  to  a—b  to  equal 
«-(/^-3)? 

42.  Write  an  expression  equal  to  a—(b—o)  without 
using  a  parenthesis. 

43.  How  much  greater  is  a  —  (ib—b)  than  a—b}  Why? 

44.  How  much  must  be  added  to  a—b  to  equal 
fl-(^-5)? 

45.  Write  an  expression  equal  to  a—(^b—b)  without 
using  a  parenthesis. 

46.  How  much  greater  is  a—{b—c)  than  a—b}    Why? 

47.  How  much  must  be  added  to  a—b  to  equal 
a-{b-c)} 

48.  Write  an  expression  equal  to  a— (<^—<:)  without 
using  a  parenthesis. 

49.  The  additive  term  b  and  the  subtractive  term  —c 
withi7i  the  parenthesis  occur  as  what  kind  of  terms  when 
the  expression  is  written  without  a  parenthesis  ? 

REMOVAL    OF    TERMS    FROM    A    PARENTHESIS    PRECEDED    DY    THE 
PLUS    SIGN. 

50.  When  the  sum  of  several  numbers  is  to  be  found, 
increasing  one  of  the  numbers  has  what  effect  on  the 
sum  ?  Decreasing  one  of  the  numbers  has  what  effect  on 
the  sum  ? 


PARENTHESIS    PRECEDED    BY    PLUS    SIGN.  2$ 

51.  How  much  less  is  a-\-{b-\-c—d—f)  than  a-\- 
{b+c-d+e-f)  ? 

52.  How  much  must  be  added  to  a-\-(^b-\-c—d—/)  to 
equal  ^  +  (^-f<:~^+^—/)  ? 

53.  Write  an  expression  equal  to  a-\-{b-\-c—d-^e—f), 
where  e  appears  outside  the  parenthesis. 

54.  Can  c  be  thus  written  outside  the  parenthesis  in- 
stead of  ^  ? 

55.  Can  a7iy  additive  term  be  thus  taken  outside  of  a 
parenthesis  ? 

56.  How  much  greater  is  a-\-(^b-\-c—d-\-e)  than  a-\- 
{b  +  c-d^e-f)} 

57.  How  much  must  be  subtracted  from  a  4-  (,b-{-c—d-^e) 
to  equal  a-\-{b-\-c—d-\-e—f')  ? 

58.  Write  an  expression  equal  to  a-\-{b-\-c—d-\-e—/)^ 
where  —/appears  outside  the  parenthesis. 

59.  Can  —d  ho.  thus  written  outside  the  parenthesis 
instead  of  —fi 

60.  Can  a7iv  subtractive  term  be  thus  written  outside 
of  a  parenthesis  ? 

61.  Can  ^wjrterm,  additive  or  subtractive,  be  taken  out 
of  a  parenthesis  ? 

62.  Can  any  tzvo  terms  be  taken  out  of  a  parenthesis 
one  after  another  ? 

63.  Can  any  number  of  terms  be  taken  out  of  a  paren- 
thesis ? 

64.  Can  all  the  terms  be  taken  out  of  a  parenthesis  ? 

21.  These  questions,  if  rightly  answered  and  under- 
stood, lead  to  the  fact,  that  when  a  parenthesis  is  preceded 
by  a  -\-  sign  the  parenthesis  viay  be  erased  and  the  value 
will  not  be  changed. 


26  REMOVAL    OF    PARENTHESES. 

REMOVAL    OF    TERMS    FROM    A    PARENTHESIS     PRECEDED    BY 
THE  MINUS  SIGN. 

22.  When  the  difference  of  two  numbers  or  expressions 
is  to  be  found,  the  number  or  expression  from  which  some- 
thing is  subtracted  is  called  the  Minuend  and  the  num- 
ber or  expression  subtracted  is  called  the  Subtrahend 
and  the  result  is  called  the  difference,  or  Remainder. 

65.  When  the  difference  of  two  numbers  is  to  be  found, 
increasing  the  subtrahend  has  what  effect  on  the  dif- 
ference ?     Why  ? 

Decreasing  the  subtrahend  has  what  effect  on  the  dif- 
ference ?     Why  ? 

66.  How  much  greater  is  a—{b-\-c—d—f^  than  a — 
{b^c-d^e-f)  ? 

67.  How  much  must  be  subtracted  from  a—{b-\-c—d—f) 
to  equal  a—{b-\-c—d-\-e—f)  ? 

68.  Write  an  expression  equal  to  a—{b  +  c—d-\-e—f)y 
where  -{-e  does  not  appear  in  the  parenthesis. 

69.  When  the  additive  term  -f  ^  is  taken  out  of  the 
parenthesis,  it  becomes  what  kind  of  a  term  ? 

70.  Can  -f<^  be  removed  from  the  parenthesis  instead 
of  -f^?  If  it  is  so  removed,  what  kind  of  a  term  does  it 
become  ? 

71.  Can  ariy  additive  term  be  thus  removed  from  a 
parenthesis  preceded  by  a  minus  sign  ? 

What  kind  of  a  term  does  it  become  when  it  is  thus 
removed  ? 

72.  How  much  less  is  a—{b-{-c—d-\-e)  than  a— 
{b+c-d-^e-f)  ? 

73.  How  much  must  be  added  to  a—i^b-^c—d-Ve)  to 
equal  a—{b-\-c—d-{-e—f)'^ 


PARENTHESIS    PRECEDED    BY    MINUS    SIGN.         2/ 

74.  Write  an  expression  equal  to  a—{b-\-c—d-\-e—f^^ 
where  the  term  —/does  not  appear  in  the  parenthesis. 

75.  When  the  subtractive  term  — /  is  removed  from 
the  parenthesis,  it  becomes  what  kind  of  a  term  ? 

76.  Can  —  ^  be  thus  removed  from  the  parenthesis  in- 
stead of  — y?  When  thus  removed,  it  becomes  what  kind 
of  a  term  ? 

77.  Can  any  subtractive  term  be  thus  removed  from  a 
parenthesis  preceded  by  a  —  sign  ? 

78.  Can  any  term,  additive  or  subtractive,  be  removed 
from  a  parenthesis  preceded  by  a  —  sign  ? 

79.  Can  any  iivo  terms  be  removed,  one  after  another, 
from  a  parenthesis  preceded  by  a  —  sign  ? 

80.  Can  any  number  of  terms  be  removed  from  a  paren- 
thesis preceded  by  a  —  sign  ? 

81.  Can  all  the  terms  be  removed  from  a  parenthesi.s 
preceded  by  a  —  sign  ? 

82.  When  all  the  terms  are  removed  from  a  parenthesis 
preceded  by  a  —  sign,  the  additive  terms  become  what 
kind  of  terms  ?  The  subtractive  terms  become  what  kind 
of  terms  ? 

23.  These  questions,  if  rightly  answ^ered  and  under- 
stood, lead  to  the  fact  that  whenever  any  parenthesis  is  pre- 
ceded by  a  —  sign,  all  the  terms  within  the  parenthesis  may 
be  taken  out,  provided  that  in  doing  so  each  additive  term 
be  changed  to  a  subtractive  term  and  each  subtractive 
term  be  changed  to  an  additive  term  ;  or,  what  comes  to 
the  same  thing,  whenever  a  parenthesis  is  preceded  by  a  — 
sigyi,  the  parenthesis  may  be  erased,  provided  all  the  terms 
within  the  parejithesis  be  changed  fro7n  additive  to  subtrac- 
tive or  from  subtractive  to  additive  as  the  case  may  be. 


28  REMOVAL    OF    PARENTHESES. 

Changing  all  the  additive  terms  of  an  expression  into 
subtractive,  and  all  the  subtractiveinto  additive,  is  spoken 
of  in  algebra  as  changing  all  the  signs  of  the  expression, 
although  it  will  be  observed  that  the  first  term  of  an  ex- 
pression has  no  sign  at  all.  While  this  latter  way  of 
speaking  is  not  quite  so  definite  as  the  former,  still  it  is 
much  shorter  and  is  very  convenient. 

24.  We  have  already  considered  the  case  of  paren- 
theses, preceded  by  the  sign  -f  and  also  by  the  sign  — , 
but  when  a  parenthesis  stands  at  the  beginning  of  an  ex- 
pression, it  is  usually  not  preceded  by  any  sign  at  all. 
This  case,  however,  is  so  simple  that  it  is  very  easily 
settled. 

In  an  expression  like 
a-{-b—c—d 
we  are  first  to  add  together 
the  numbers  a  and  b,  and 
from  this  sum  subtract  c, 
and  then  from  this  differ- 
ence subtract  d. 


In  an  expression  like 
{a  +  b—c)—d 
we  are  first  to  add  together 
the  numbers  a  and  b,  and 
from  this  sum  subtract  c, 
and  then  from  this  differ- 
ence subtract  d. 

From  these  two  statements,  which  are  exactly  alike, 
it  is  evident  that  when  a  parenthesis  stands  at  the  begin- 
ning of  an  expression,  the  expression  means  exactly  the 
same  thing  that  it  would  if  the  parenthesis  were  erased, 
and  from  this  it  is  evident  that  when  a  parenthesis  is  pre- 
ceded by  7io  sign  at  all,  the  parenthesis  may  be  erased,  just 
the  same  as  though  the  parenthesis  were  preceded  by  the 
sign  +,  or  in  other  words,  when  a  parenthesis  is  pre- 
ceded by  no  sign  at  all,  it  is  treated  just  as  though  it 
were  preceded  by  the  sign  +. 


MISCELLANEOUS    EXAMPLES. 


29 


EXERCISE    1  1. 

Miscellaneous  Examples  on  the  Removal  of  Parentheses. 

Write  each  of  the  following  expressions  without  using 
any  parentheses  : 

1.  .;,7  +  /^^-(/-f^^-r).  4.   (^a^b-^c)-{q-r). 

2.  m  +  n+p-{q  +  r).  5.   a^{a''-b)-{b'' +c). 

3.  {a-b)  +  {p-q)-\0.  6.   a-b-^a'' ■^b--c''). 

8.  (^ab-a)-\-{;dab'' -2cd)~(cd'^ -a"-). 

9.  {Am-2n)  —  (?,p-2q  +  r). 
10.   m  —  {a-\-b)  —  {c—d). 

12.  (w--«2)-(/>+2^)  +  (6r2-fa/^-^3). 

13.  (a2_^^2^«^)-(^z;+5/^r-^/^r). 

14.  (lax— ^by)  —  {^ax— by')  — 10. 

15.  C)a'^x-(lZby'^—babxy'-''dy^)^(Jx^-^y'^). 

16.  21-(5-8y+13>/)-(6  +  16j'-15,r). 

17.  75-(15-8)/+2l7)-(30+80>/-lo.v2). 

18.  (3«a--5y)-C2«y-|.r)  +  r75'-3)0. 


CHAPTER  III. 
ADDITION. 

EXERCISE  12. 

Addition  of  Expressions. 

25.  When  two  or  more  expressions  are  to  be  added 
together,. each  expression  may  be  enclosed  in  a  parenthe- 
sis, these  parentheses  written  one  after  another,  separated 
by  plus  signs.     Thus,  if 

a -^2,     Oil — 1    and   a-\-3 
are  to  be  added,  w^e  would  indicate  the  sum  by 

(a  +  2)  +  (3«-l)  +  (^4-3). 
These  parentheses  ma}^  now  be  removed,  and  the  sum  in- 
dicated thus  :  «  +  3a  +  a  +  2  — 1  +  3 
which,  by  uniting  the  terms,  may  be  written 

5^  +  4. 
I^et  us  find  the  sum  of  the  three  expressions 
x-\-y,    x—z   and    "Ix-^-Zy—^, 
First,  we  enclose  these  in  parentheses,  write  them  one 
after  another  separated  by  plus  signs,  and  get 

C-^+j)  +  {x-z)  +  (2.r +  3j/-2). 
Second,  we  remove  parentheses  as  in  exercise  10,  and  get 

x^-y-\-x—z^2x-\-?>y—2. 
Third,  we  arrange  these  terms  so  that  similar  terms  shall 
come  together,  and  get 

x-\-x^1x-\-y-V'^y—z—1. 
Fourth,  we  unite  each  group  of  similar  terms  into  a  single 
term  and  get  4x-{-4y—z'—2, 

and  this  is  the  simplest  form  possible  for  the  sum  of  the 
three  given  expressions. 


ADDITION    OF    EXPRESSIONS.  3 1 

26.  It  is  easy  to  see  that  we  could  find  the  sum  of 
any  number  of  expressions,  whatever  those  expressions 
may  be,  in  a  manner  similar  to  that  just  pursued,  viz. :  en- 
close each  expression  in  a  parenthesis,  write  these  paren- 
theses one  after  another  separated  by  plus  signs,  and  then 
remove  parentheses  from  the  expression.  Next  arrange 
the  terms  of  the  expression  thus  found,  so  that  similar 
terms  shall  come  together,  and  then  by  uniting  each 
group  of  similar  terms  into  a  single  term,  we  obtain  the 
simplest  form  possible  for  the  required  sum  of  the  given 
expressions. 

EXAMPLES. 

Find  the  sum  of  the  following  expressions  : 

1.  x—?yy—4,     2x—ijj'--(j  and  ;«r-f  3y+12. 

2.  3/2--2«<^-f5/^2^    a^—Aad—3d^   and  a^-\-d'\ 

3.  4jry-_>^2_}.23,     6jrv+2y_50  and  Sy--xy-2. 

4.  Jt:2-f2.rj'4-r-,    x^--2xy-hj'''   and  x--j\ 

5.  Gw— 4^/— r2,     2^/+2w-fG?'2    and   li)s^—7r^-—m. 

6.  3jr2+^j-f3>/2,    x'^-Sxj+y-   and  ?>x''-\-^j\ 

7.  a  +  b,     12c-j-Sd,    Qd-4d  and  lOa-^d. 

8.  a-i-U,     d+4c,     c+4d  and  d-h4e. 

g.  oxy-\-2y,     hxy—x,     Sx—iry  and   7xy—x—2y. 

10.  Sa—4d—ecd+2e,   10/^+3^— lOr^  and  9a —20^4- ^W. 

11.  (yc-{-7  —  4a—5d,    ijd-\-7c—4—5a,  and  Ga  +  7d—4c-o. 

12.  a^-^-P,    a^-d^  and  ad^-+aH-\-d\ 

EXERCISE  13. 

Arrangement  of  Work  in  Addition. 

27.  In  finding  the  sum  of  two  simple  expressions  by 
the  method  already  learned,  we  place  these  expressions 
in  parentheses,  writing  them  one  after  another,  separating 


32  ADDITION. 

them  by  plus  signs.  But  evidently  it  comes  to  the  same 
thing  if,  instead  of  writing  the  expressions  one  after  an- 
other within  parentheses,  we  write  them  one  below  another 
without  parentheses,  arranging  the  similar  terms  in  the 
same  vertical  column.  We  may  then  draw  a  line  under 
the  last  expression,  and  the  example  is  arranged  in  ex- 
actly the  same  form  as  in  Arithmetic.  Now  the  similar 
terms  in  each  column  may  be  combined  into  a  single  term, 
as  we  alread}^  know,  and  this  term  placed  under  the  line 
as  one  term  of  the  sum.  When  all  the  columns  are  thus 
treated  all  the  terms  of  the  sum  are  found.  Thus,  sup- 
pose we  are  required  to  find  the  sum  of 

^a'^-^W-hcd,    6«2_2^2  and  W-W-A^cd. 
By  the  former  method  we  write 

Removing  parentheses  we  get 

2a2+3^2_5^^^(3^2_2^2_|.§^2_4^2_4^^^ 

Arranging  similar  terms  one  after  another  we  get 

2a2^e)^2_4^2_^3^2_2^2^8^2_5^^^_4^^^ 

Uniting  similar  terms,  we  get 

By  the  latter  method  we  write 

2a2  +  3^2_5^^ 
6a2-2^2 
--4a2  +  8^2_4^^ 
4^2_^9^2_l9^^ 

Now  it  is  very  evident  that  we  have  here  exactly  the 
same  terms  to  combine  that  we  had  by  the  other  method, 
after  the  parentheses  had  been  removed,  and  the  similar 
terms  brought  together.  Moreover,  it  is  apparent  that 
these  terms  are  combined  in  exactly  the  same  order,  giving 


EXAMPLES.  33 

of  conrre  the  same  result  as  before;  and  the  only  dif- 
ference between  the  two  methods  consists  in  the  arrange- 
ment of  the  work.  The  similar  terms  being  rather  easier 
to  combine  in  the  second  arrangement,  it  is  the  more  con- 
venient arrangement  for  the  beginner. 

EXERCISE  14. 

Examples. 

I.  2. 

«2-f  ab-^  b"-  5ab-\-Gbc—7ca 

W-?yab-lb''  ^ab-Abc+Zca 

ia^+nab-^db^  2ab—  bc-^bca 

3-  4. 

2x'^  —  '2xy-\-^y'^  hax—lby-{-  cy 

Sx^--\-hxy-\-4y^  Sax-\-dby-^cy 

x'^—2xy—^)y'^  ax—^by—  cy 

5.  Find  the  sum  of  5/;-f4/+3?^,  2/^-}-2/^-^^,  and 
7/^  +  3/+ 4/^. 

If  //  be  supposed  equal  to  100,  /  equal  to  10  and  u  equal  to  unity, 
this  example  illustrates  nicely  the  analogy  between  the  work  the  stu- 
dent is  now  doing  and  the  ordinary  addition  of  numbers  in  Arithmetic. 

Thus, 

Hundreds.    Tens.     Units. 

5^4-4/-f3«  5            4            3 

2//-J-2/-I-  w  2            2            1 

1h\-V-\.\u  n_ 3 4 

14//+9/+8«  14             9             8 

6.  Arrange  and  add   x-hy+^,     2x-\-?yy—22,    and   3a- 

7.  Add  14«-6^-h3^~5arand  S)a  +  lb-Ac-S)d. 

8.  KM  a-\-b—c-^d  Tind  a—b—c^?yd. 

9.  Add  ba'^-Zab  +  lb'',  la'^^iab-bb'' ,  and  a''~S)ab 
+  'ib\ 


34  ADDITION. 

10.  Add  a-\-2d-\-oc,    2a—d—2c,    b—a^c,  and  c^a  —  h. 

Arrange  thus:  «-f-2<^-|-3^ 

la—  b—1c 

_    «_|_    /;_    c 

_  a-  /,-!-  c 

11.  Add  6/5+8r— 5«,     8«— 3/;+4r,  and  7Z^— 15r— 2^. 

12.  Add   3;/-f  2r+3^— 4/,    3r— 4^— 5/— 2«,   and   5^—0/ 
^-12;^-10r. 

13.  Add  x-\-Za-^2b-c,      2j-Sl?-\-2c+a,    and     3^+3^ 
-2a-d. 

3a-\-2/'-  c-\-x 
a-'6b-\-1c       +2y 
-la-  b^?yc  4-32 

14.  Add  3ji:3-4jt;2-.r+7,       2;r3+Jt:2-f 3;tr-10,       2;tr2 
-7;t'3-2ji;-14,  and  Zx^-\2x''A-\2^-hx. 

In  arranging  expressions  which  involve  different  powers  of  the 
same  number,  it  is  usual  to  place  all  of  the  terms  containing  the 
highest  power  of  the  letter  in  the  first  column,  all  the  terms  con- 
taining the  next  highest  power  in  the  next  column,  and  so  on.  Thus, 
this  example  would  generally  be  arranged  in  this  way: 
S-r'—  4jc8—  x-\-  7 

-7^3_j_  2jc2-2-jr-14 

15.  Add    .r3+4jt:2  +  5.r-3,      2x^—7x'^  —  Ux-h5,    and 
^2_^3_2  +  i0jr. 

16.  Add  Sx^—ix^-^x*,     x^-{-x'^+x,     4x^-i-5x\  and 
2x''-Sx-4:x\ 

17.  Add  5x^-Sx^-hZx-8,      x^--Sx^-\-Sl,      2x^-8x 
-5.^2+2,  and  ix+lx^-d-SxK 

18.  Add  Sx^-4xj^-\-y^-\-2x-\-Sy,     lOxj-i-Sy^+dj^,  and 
bx''-Qxy+Sy^-\-Jx-7j^. 

19.  Add    4ad^+icde-6/i\       2ad'--\-3/i^,      dcde-7ad\ 
and   ab''-^2cde-'5/i^. 


EXAMPLES.  35 

20.  Add   ^xy-V^ode-lfg,      S/g-2xy,      Sxj-Sde,      Sde 
~-^xy—2fg,  and  ^/g—2xy. 

21.  Add  U'^a^-1aH+^ab''-^l^b^  2inA  1\a^-2\a''b 
-4ab''-12b\ 

22.  Add  9ir+3f^+7.75  and  7ic-Uj%d-S. 

23.  Add|«  +  ^,     ib-ia,     ^aSb,  and  3^-9^^. 

24.  Add   a^+SaH-{-oab'^-\-b\     a^-3aH-hSab^-b^, 
b^-3ab''-i-SaH-a'\  and  a^-h6aH  +  Gab''-\~bK 

25.  Add     12j/—5a—7ax,       5ax-\-a—Sy,       da—y—ax, 
4ax—Sa-i-5y,  and  y-\-a—ax. 

26.  Add    «3^5«/^2_^/!i3^      a^-10ab''-hb\     and     5a^2 

27.  Add   x^—2ax'^+a'^x-^a^,      x^-^Sax^,     and     2«3 
—ax'^—x^. 

28.  Show    that    if   x=a  +  2b—Sc,    y=b-\-2c—Sa,    and 
^=f+2rt— 3^,  then  will  x+y+z=0. 

29.  Add  3«4-2^— <:,     Sb-\-2c-a,  and  3^+2a— ^. 

30.  Add  2^-3^24.^2^     p^2c^  +  Sd,  and  3^3-2^-^. 


CHAPTER  IV. 

SUBTRACTION. 

EXERCISE  15. 

Subtraction  of  Expressions. 

28.  When  one  expression  is  to  be  subtracted  from 
another,  we  may  enclose  each  expression  in  a  parenthesis 
and  separate  the  minuend  from  the  subtrahend  by  a  minus 
sign.  Thus,  if  «  +  2  is  to  be  subtracted  from  Sa—1,  we 
would  indicate  the  difference  by 

(3«-l)-(«  +  2). 
These  parentheses  may  now  be  removed,  care  being  taken 
that  all  the  signs  in  the  second  parenthesis  be  changed 
when  the  parenthesis   is  removed,   and   the    remainder 
written  thus,  3^—1—^—2. 

Now  by  grouping  the  terms,  the  same  result  may  be 
written  3^—^—1—2, 

which  by  uniting  the  terms  may  b*^  written 

2^-3. 
which  is  the  required  difference. 

Let  us  find  the  difference  between 

Sx''-4:xy+5y^  and  2x'^-2x_y-4j;\ 

First,  we  enclose  each  of  these  expressions  in  a  paren- 
thesis, writing  the  subtrahend  after  the  minuend,  with  a 
minus  sign  between  them,  and  get 

(Sx^-4:Xj^-\-5j/'')-i2x''-2xy-4y). 

Second,  we  remove  each  of  these  parentheses,  taking 
care  to  change  all  the  signs  within  the  second  parenthe- 
sis, and  get 

dx^-4xy-j-5y-'2x'^-{-2xy-\-4y. 


SUBTRACTION    OF    EXPRESSIONS.  37 

Third,  we  arrange  these  terms  so  that  similar  terms 
shall  come  together,  and  get 

Fourth,  we  unite  each  group  of  similar  terms  into  a 
single  term,  and  get 

and  this  is  the  simplest  form  possible  for  the  difference  of 
the  two  given  expressions. 

It  is  easy  to  see  that  we  could  find  the  difference  of 
any  two  expressions,  whatever  these  expressions  may  be, 
in  a  manner  similar  to  that  just  pursued,  namely,  enclose 
each  expression  in  a  parenthesis,  write  the  minuend  first 
and  the  subtrahend  second,  with  a  minus  sign  between 
them,  and  then  remove  parentheses.  Next  arrange  the 
terms  of  the  expression  thus  found  so  that  similar  terms 
shall  come  together,  and,  finally,  by  uniting  each  group 
of  similar  terms  into  a  single  term  we  obtain  the  differ- 
ence of  the  two  given  expressions  in  the  simplest  form 

possible. 

EXAMPLES. 

1.  From  17a-\-id—?>ctake  ?>a-\-od—c. 

2.  From  ^x-dy  +  20^  take  jir-4>'+3^. 

3.  What  is  the  difference  between  (jd-\-25c  and  Qd—25c? 

4.  What  is  the  difference  between  8.r ^  H-  4j  and  Sx"^  —iy} 

5 .  From  X-+  2xy -\-y -  take  x -  —  2xy -\-y ^ . 

6.  From4jf2  +  2;t:j4-o>'2  tixk^  x"^ - xy -\-2y'^ . 

7.  From  6j»;2-13a'-7  take  3.r2_^5jr-2. 

8.  From  2;»;— 11«  +  10^— 5^— 23  take  ba-^2c—10—U. 

9.  From  2A-'^+ji;2-35;r+49  take  x^-2^x+A2. 

10.  From  4.x''-^x^-2x''+lx+%  take  x"^ -2x'' -2x'^ 
■~7a--9. 

11.  From  a^-\-?>a''b^?^ab"'-\-b^  i2ikQ  a^-^a'^b+Zab'^-b^ 

12.  From  3;r+10  take  10— 3j/. 


38  SUBTRACTION. 

t 
EXERCISE   16. 

Arrangement  of  Work  in  Subtraction. 

29.  In  finding  the  difference  of  two  expressions  by 
the  method  ah'eady  learned,  we  place  each  expression  in 
a  parenthesis,  writing  the  subtrahend  after  the  minuend 
with  a  minus  sign  between  them,  and  then  remove  the 
parentheses.  But,  evidently,  it  comes  to  the  same  thing 
if,  instead  of  writing  the  subtrahend  with  all  its  signs 
changed  after  the  minuend,  we  write  the  subtrahend 
with  all  its  signs  changed  below  the  minuend,  placing 
similar  terms  of  the  minuend  and  subtrahend  in  the  same 
veritical  column,  and  then  uniting  similar  terms  exactly 
as  in  addition.     For  example,  if  we  wish  to  subtract 

we  arrange  the  work  thus  : 

Minuend  9^2H-332_7 

Subtrahend  with  signs  changed  — 2^2+4^^ -f  6 
Remainder  1  a'^ -\-l  b'^ —  \ 

The   signs    of  the    subtrahend   need   not   actually  be 

changed  if  the  student  will  iraagine  them  changed  as  he 

proceeds  in  the  work. 

EXERCISE  17. 

Examples. 
I.  2. 

From     6:r-14ji/4-10  9^-43+8^ 

take        1x—  ^y-\-   6  a-^1b^2,c 

3-  4. 

From     9«— 8^4-7<r— 3^  1x—2y-{-Zz—\u 

take      5^—6^— 3r+2^  hx-V^^y—hz^^tc 


EXAMPLES.  39 

5.  6. 

From     4x—Sjy-\-du^Sv  m—Sfi-j-p—7 

take       2x-h4y—Su—Sv  m—4fi—p-\-8 

7.  8. 

From     ^a-2i^-\-S^c  a-Sd-\-4c-2d 

take       l^^  +  H^+4|f  7i?-2c-Sd+e 

g.  10. 

From     li')a—7d+oc—7cf—Se  7x—2j'—  :r-\-4-\-a 

take       \0a-\-7b—?>c-\-4d^4e  x+  y-{-hz—2       -\-n 

II. 

From     7?ya  —  r^2b—7\c-\-2\d—:)2x+\7y-\-r)^dz+n 
take       54^— G0<^+81r+::]7^-f-18A-— :%>-t-99^-f   7 

12.  From  4x'^-\-2xj'+?>v'  take  Jt-— ji:)/+2:)'2. 

13.  From  rt»  +  3a2^+3«/^2_|.^3take«'^-3rt2^-f  3«^--/^'' 

14.  From  2A-f  ll^  +  10Z-5r-23  take  2^-10+5^-3^. 

15.  From  ?>x^-2x'--\-?>x—4  take  .^'3-4J»:2-8A-f  1. 

16.  From  72;r-*-78A-'^-10.r'  +  17  take  2^-*+ 30x^-1 7.r 
+  10.r2. 

17.  From  3;»:'*+5A'*^-6.r2-7.r+5  take  2x*-2x''-h5x^ 
— Gjt- 7. 

18.  From  7;t:2— 8j»:— 1  take  5^-2— 6.r+3+.r2. 

19.  From4Ar*-3.r»-2A-2  — 7jt:+9    take   .r* -2.^^-2.^-2 
+  7.r-9. 

20 .  From  5.^ 2  _^ g^^, _  1 2 jr^  —  4j' 2     ^ake    2;i- 2  _ 7^^, _j_ 4^^, 

21.  From  x^-\-Sxjy—j'^  ■^yz—2y'^    take   x"^ -{■  2xy  +  5;r.^ 

22.  From  7ji:*— 2.^2  +  2^-4-2  take  4jr3  — 2ji:2— 2.^-— 14. 

23.  From4.r3— 2.;r2— 2;*:— 14^  take  2jt:-'^— 8j»:2+4A-+a^. 


40  SUBTRACTION. 

24.  From  3i«-24/!i 4-31^- 5^  take  na  +  U-'i\c+Sd. 

25.  From  (.)ay—5xy+2a'^x^  take  4jr;'— 3^) — a'^x-. 

26.  From  J^  +  J-^-f^-ll^+i  take4cz-|^— V-^+W-l 

27.  From  ^  +  2  take  d—o. 

28.  From  «  +  <^  take  a-i-c. 

29.  From  ^  +  ^  take  c-{-d. 

30.  From  a  — <^  take  d—c. 

31.  From  «"— ^^4-<^2  take  Jtr2  4-^_)/+_>/2_ 

32.  From  4al;y-  —  5axy-^2a'^x"  take  «2;t:2_^^^_3^^^2^ 

33 .  From  6x'^-\- 7xj —oy'^  —  l 2xy2 — Sjz   take    8xy — lyz 
-\-Sx^—4y'^-\-6xy2. 

34.  From  2x-^na  +  10d-rDc-2^  take  2c-10  +  5a-Hd. 

35.  From  4r^+62;^'^-26;;^-23;^2    take    d?i''-2rs  +  21m 
•j-2n''-.Srs. 

EXERCISE   18. 
Additon  and  Subtraction  of  Equals. 

1.  How  much  less  is  x+i  than  ^+5?  Ifjr4-5=12, 
what  does  x-\-4  equal  ?  Why  ?  What  does  x-^S  equal  ? 
Why  ?  What  does  x-\-2  equal  ?  What  does  x-\~l  equal  ? 
What  does  x  equal  ? 

2.  If  jt:+5=28,  what  does -r+3  equal  ?^  Why?  What 
does  x+1  equal  ?     Why  ?     What  does  x  equal  ?     Why  ? 

3.  If  ^+30=50,  what  does  ;r+10  equal?  Why? 
What  does  x-\-7  equal  ?  Why  ?  What  does  x+o  equal  ? 
Why  ?     What  does  x  equal  ?     Why  ? 

4.  If  jtrH- 12=44,  write  what  x  equals.  What  must  you 
do  to  each  member  of  the  equation  ^+12=44  to  get  the 
equation  ;«;=32  ? 

5.  If  ;r+ 23=48,  write  what  x  equals.  What  must  you 
do  to  each  member  of  the  equation  ji;+23=48  to  get  the 
equation  you  have  just  written  ? 


ADDITION   AND    SUBTRACTION    OF    EQUALS.       4I 

6.  If  x-^27i—o7i,  write  what  x  equals.  What  must  you 
do  to  each  member  of  the  equation  x+2n=6?i  to  get  the 
equation  you  have  just  written  ? 

7.  If  x-\-n=4:5,  write  what  x  equals.  What  must  you 
do  to  each  member  of  the  equation  x-\-7t=4o  to  get  the 
equation  you  have  just  written  ? 

8.  If  x-\-?i—a,  write  what -T  equals.  What  must  3^ou 
do  to  each  member  of  the  equation  x-^7i  =  a  to  get  the 
equation  you  have  just  written  ? 

30.  In  finding  the  value  of  x  in  the  above  equations 
the  student  has  made  use  of  a  ver}-  evident  mathematical 
truth,  which  may  be  stated  as  follows  : 

If  we  take  from  equals  the  same  niunber,  or  equal  num- 
bers, the  remainders  will  be  equal. 

This  is  one  of  several  equallj^  evident  truths  which  are 
known  in  mathematics  as  Axioms. 

9.  How  much  more  is  .r— 4  than  x—b}  If  x—b=?), 
what  does  x—\  equal  ?  Why  ?  What  does  ;r— 3  equal  ? 
Why  ?  What  does  x—2  equal  ?  Why  ?  What  does  .r-1 
equal  ?     Why  ?     What  does  x  equal  ?     Why  ? 

10.  If  ;r— 7  =  8,  what  does  x—b  equal?  Why?  What 
does  ji'— 3  equal  ?     Why  ?     What  does  x  equal?     Why  ? 

11.  If  ;«;— 20=25,  what  does  x—lb  equal?  Why? 
What  does  x—\0  equal?  Why?  What  does  x—4: 
equal  ?     Why  ?     What  does  x  equal  ?     Why  ? 

12.  If  .r— 11  =  13,  write  what  x  equals.  What  must 
you  do  to  each  member  of  the  equation  .r— 11  =  13  to  get 
the  equation  .r=24  ? 

13.  If  JT— 16=50,  write  what  x  equals.  What  must 
you  do  to  each  member  of  the  equation  ;r— 16=50  to  get 
the  equation  you  have  just  written? 


42  SUBTRACTION. 

14.  If  xSn=5n,  write  what  x  equals.  What  must 
you  do  to  each  member  of  the  equation  x—S?i=5ji  to  get 
the  equation  you  have  just  written? 

15.  If  x—n=15,  write  what  x  equals.  What  must 
you  do  to  each  member  of  the  equation  x—7i  =  15  to  get 
the  equation  you  have  just  written? 

16.  If  x—7i=a,  write  what  x  equals.  What  mUvSt  you 
do  to  each  member  of  the  equation  x—7i=a  to  get  the 
equation  j^ou  have  just  written  ? 

31.  In  finding  the  value  of  x  in  the  above  equations 
the  student  has  made  use  of  another  very  evident  mathe- 
matical truth,  or  axiom,  which  is  usually  stated  as  follows: 

If  we  add- to  equals  the  same  number  or  equal  7iumbers^ 
the  SU7US  will  be  equal. 

EXERCISE  19. 

EXAMPLES. 

Find  the  value  oi  x  in  each  of  the  following  equations, 
by  the  addition  or  subtraction  of  equals : 

1.  ^_i9=32. 

We  have  x- 19  =  32 

Adding  equals  to  each  member  19     19 

"Whence  x=51 

2.  5^+12  =  87. 

We  have  5-r-|-12  =  87 

Subtracting  equals  from  each  member  12     12  ' 


bx—1b 

Whence  we  know 

x=Vo. 

3.  x-^  =  ri. 

7. 

5.r+3=2S. 

4.  jt-+17=19. 

8. 

7x+5=26. 

5.  A-+21  =  69. 

9. 

5  +  6jr=29. 

6.  A-19=43. 

10. 

9.r-5=31. 

TRANSPOSITION    IN    EQUATIONS.  43 

11.  ;i;4-140=191.  17.  194=ll;»;-26. 

12.  18+^=35.  18.  bx-\-2-hx=20. 

13.  ^—48=56.  19.  4ji:+54-7^+9— 8:r=16. 

14.  25+x=38.  20.  dx-hl2-6x-l^-{-2x=ld, 

15.  2=;r-8.  21.  9A-+l?)-;t-=29. 

16.  230=;»;-103.  22.  13:r+9-8a-=39. 

EXERCISE  20. 

Transposition  in  Equations 

1.  In  the  following,  explain  how  each  equation  in  the 
second  column  can  be  obtained  from  the  corresponding 
one  in  the  first  column  : 

(1)  x-h4=d.  (V)  .r=9-4. 

(2)  x+2c==5c,  (2')  x=5c-2c. 

(3)  x-\-2c=12.  (3')  x=12-2c. 

(4)  x+a==d.  (4')  x=:d-a. 

(5)  jt:-6=3.  (5')  x^n-i-i). 

(6)  x-2d=?yd.  (6')  x==Sd+2d. 

(7)  x-2d^l.  (7')  .r=.7  +  2^. 

(8)  x-a^b.  (8')  ji-=^-f^. 

2.  The  additive  terms  -|-4,  -f  2r,  and  +«  from  the  /r/? 
members  of  the  equations  in  the  first  column  appear  as 
what  kind  of  terms  in  the  right  members  of  the  equations 
in  the  second  column  ? 

3.  The  subtractive  terms  —6,  —2d,  and  —aoi  the  left 
members  of  the  equations  in  the  first  column  appear  as 
what  kind  of  terms  in  the  right  members  of  the  equations 
in  the  second  column  ? 

32.  From  the  above  work  we  learn  the  following 
principle  : 


44  SUBTRACTION. 

By  the  addition  or  subtraction  of  equals,  we  may  cause  a 
terin  to  disappear  from  any  member  of  an  eqtiation  and  to 
appear  ivith  its  sign  changed  in  the  other  7nember  of  the 
equation. 

If  we  remove  a  term  from  one  member  of  an  equation 
and  make  it  appear  in  the  other  member,  we  are  said  to 
Transpose  that  term.  If  we  nse  this  word  we  maj^  re- 
state the  above  principle  as  follows  : 

A7ty  term  in  07ie  me7nber  of  a7i  equation  may  be  trans- 
posed to  the  other  7ne77iber  provided  its  sign  be  changed. 

In  this  way  of  speaking,  we  are  apt  to  keep  in  mind  merely  the 
change  which  results  in  the  equation  and  to  lose  sight  of  the  addition 
or  subtraction  of  equals  which  causes  transposition.  The  axioms  must 
always  be  appealed  to  when  we  are  called  upon  to  explain  why  trans- 
position is  allowable. 

EXERCISE  21. 

Examples. 

33.  To  Solve  an  equation  is  to  find  the  value  of  the 
unknown  number  in  the  equation,  and  the  process  of 
finding  this  unknown  number  is  called  the  Solution  of 
of  the  equation. 

As  mistakes  may  be  made  in  the  solution  of  an  equa- 
tion, it  is  well  for  the  student  to  test  the  results  found,  by 
putting  in  the  original  equation  the  value  obtained  for 
the  unknow^n  number  in  place  of  the  letter  representing 
it.  If  the  equation  thus  found  is  not  true,  a  mistake  has 
been  made  and  the  solution  should  be  re-examined. 

This  process  of  testing  a  result  is  called  the  Verifica- 
tion of  that  result. 

Solve  each  of  the  following  equations  : 
I.  5ji;— 2=3.r-f  18. 


EXAMPLES.  45 

SOLUTION. 

Transposing  the  terms  'dx  and  —2, 

5x-Sx=lS-\-2. 
Uniting  terms,  2^=20; 

whence  jr=10. 

VERIFICATION. 

Substituting  10  for  x  in  the  original  equation, 
5X10-2  =  3X10-1-18, 
or  50-2  =  30-1-18; 

and,  since  this  equation  is  true,  the  correct  value  of  x  has  been  found- 

2.  30  +  5jt:=70.  4.   7j»:-8=41. 

3.  9;«:-5=31.  5.   1oa'-13=107. 

6.  28-8r=7. 

SOLUTION. 

Transposing  the  terms  —  3t'  and  7, 

28-7=3/. 
Uniting  terms,  21  =  3/; 

whence  /=7. 

VERIFICATION. 

Substituting  7  for/  in  the  original  equation, 
28-21  =  7. 

7.  i9-r)jt-=4.  20.  5A-+18=3jt-+3S. 

8.  107  — 13a-=42.  21.  r>Qx—27=47x. 

g.  76-19.^=0.  22.  9A'  +  17=102-8jt;. 

10.  25G-32j/=0.  23.  bx-5==2x-^?y. 

11.  8.r-f-7— -r=14.  24.  lx-\-2^Ax^l. 

12.  9.v-f  13--r=29.  25.  3.r-l  =  ll-.r. 

13.  i7^_j-i9_2v=64.  26.  .r4-4=10-2.r. 

14.  23jr— 18-f3j»:=8G.  27.  31  — 7.-r=41  — 8.r. 

15.  3ji:+18=5x.  28.  38-2j'=9j/-39. 

16.  19.r— 14=12.r.  29.  29a— 57=l()-r-5. 

17.  19;r=l()  +  lLr.  30.  147  — 19j»;=122-14a\ 

18.  7.r=10()  +  3-r.  31-  14jt:-f  23=19,r-2. 

19.  30.r=80-10ji;.  32.  50  +  17ji-=295-18.ar 


46  SUBTRACTION. 

33.  5j*r-f  13-2jr=100-20jr-18. 

34.  16:^+10— 21;r=45  —  10jr— 15. 

35.  o=U+x-8x-Sx-\-4-\-x. 

36.  7—ox—10-\-Sx—7-\-Sx=x,   • 

37.  2x-ix=U-i-ix-h2. 

EXERCISE  22. 

Problems. 

1.  What  number  increased  by  57  is  equal  to  98  ? 

Let  X  represent  the  number. 

Then,  because  the  number  increased  by  57  equals  93,  therefore, 

Transposing,  jf=93— 57. 

Uniting  similar  terms,  x  — 36. 

2.  What  number  diminished  by  26  is  equal  to  29  ? 

3.  What  number  increased  b}^  17  is  equal  to  35  ? 

4.  What  number  diminished  by  19  is  equal  to  15  ? 

5.  If  twice  a  certain  number  be  increased  by  12,  the 
result  will  be  30.    What  is  the  number  ? 

Let  X  represent  the  number. 
Then,  because  twice  the  number  increased  by  12  equals  30,  therefore, 
2.v-fl2  =  30,     etc. 

6.  If  five  times  a  certain  number  be  diminished  by  15, 
the  result  will  be  45.     What  is  the  number  ? 

7.  Five  times  a  certain  number  exceeds  three  times 
that  number  by  22.     What  is  the  number? 

Let  X  equal  the  number. 

Then,  since  5  times  the  number  exceeds  3  times  the  number  by  22, 
therefore,  5.^:— 3-^=22,  /    etc. 

8.  If   three  times  a  certain  number  be  added  to  the 
number  itself,  the  sum  will  be  36.     What  is  the  number  ? 


PROBLEMS.  47 

9.  If  five  times  a  certain  number  be  added  to  eight 
times  that  number,  the  sum  will  be  78.  What  is  the 
number  ? 

10.  If  three  times  a  certain  number  be  increased  by  5, 
the  result  is  equal  to  the  number  increased  by  25.  What 
is  the  number  ? 

Let  x=the  number. 

Then  uX-\-  5=ii  times  the  number  increased  by  5, 

and  x-|-25=the  number  increased  by  25. 

Then,  because  3  times  the  number  increased  by  5  equals  the  number 
increased  by  25,  therefore, 

3x-f5=.r-f25,       etc. 

11.  If  three  times  a  certain  number  be  diminished  by 
4,  the  result  is  equal  to  the  number  increased  by  6.  What 
is  the  number  ? 

12.  What  number  must  be  added  to  787,  so  as  to  ob- 
tain the  same  result  as  when  the  number  is  taken  from 
875? 

13.  What  number  is  as  much  greater  than  78  as  it  is 
less  than  108  ? 

14.  What  number  gives,  when  doubled,  7  more  than 
three  times  itself  diminished  by  16  ? 

15.  Having  $78,  I  spent  an  amount  such  that  I  had 
left  5  times  as  much  as  I  spent.  How  much  did  I  spend  ? 

16.  A,  B,  and  C  together  put  $8781  into  a  business. 
B  put  in  $1000  more  than  A,  and  C  put  in  $2000  more 
than  A.     Find  how  much  money  each  man  put  in. 

Let  a-=number  of  dollars  A  furnished. 

Then  .r-}-1000=number  of  dollars  B  furnished, 

and  j:-f"2000  =  number  of  dollars  C  furnished, 

and  since  altogether  they  furnished  $8781,  therefore, 

x-\-{x-\- 1 000)-h(  jc-f  2000)  =r  8781. 
Removing  parentheses  and  uniting  similar  terms 
iU-l- 3000- 8781,       etc. 


48  SUBTRACTION. 

17.  A  boat' went  368  miles  in  three  days.  The  second 
day  it  sailed  32  miles  further  than  it  did  the  first  day, 
but  the  third  day  it  sailed  24  miles  less  than  it  did  the 
first  day.     How  far  did  the  boat  sail  each  day  ? 

18.  John,  Fred,  and  Dave  have  $4.35.  Fred  has  twice 
as  much  as  John,  and  Dave  has  75  cents  more  than  John. 
How  much  has  each  ? 

19.  A  man  dying  left  his  property  worth  $52800  to 
his  four  children.  He  gave  his  oldest  child  $3000  more 
than  the  youngest,  the  next  $2000  more  than  the  youngest, 
and  the  next  $1000  more  than  the  youngest.  How  much 
did  each  receive  ? 

20.  A  merchant  made  $4222  in  three  years.  He  made 
$1592  more  the  second  year  than  he  did  the  first,  but  the 
third  he  lost  all  he  made  the  first  year  and  $350  more. 
What  did  he  make  each  year  ? 

21.  If  G3  be  added  to  a  certain  number,  the  number 
becomes  10  times  as  large.     What  is  the  number  ? 

22.  A  and  B  had  equal  sums  of  money.  A  doubled 
his  mone\^  and  then  made  $25,  while  B  tripled  his  money 
and  then  lost  $100.  They  then  had  equal  amounts. 
What  sum  did  each  have  at  first  ? 

Let  X— the  number  of  dollars  each  had  at  first. 

Then,  because  A  doubled  his  money,  and  also  made  $25,  therefore, 

2-*"-f-  2o=:the  number  of  dollars  A  had  finally. 
Also,  because  B  tripled  his  money  and  also  lost  $100,  therefore, 

3^—100 -the  number  of  dollars  B  had  finally. 
Now,  since  A  and  B  finally  had  equal  amounts 

3.r-100:=2x+25,        etc. 

23.  If  10  times  a  certain  number  be  diminished  by  22, 
there  results  the  same  as  when  7  times  the  number  is  in- 
creased by  23.     What  is  this  number  ? 


PROBLEMS.  49 

24.  A  man  doubled  the  money  he  had,  and  then  made 
$500.  He  then  made  an  amount  equal  to  3  times  what 
he  had  at  first,  but  losing  $5400  he  had  nothing  left. 
How  much  did  he  have  at  first? 

25.  A  man  walked  47  miles  in  three  days.  He  walked 
8  miles  more  the  second  day  than  he  did  the  first,  and  10 
miles  more  the  third  day  than  he  did  the  second.  Find 
how  far  he  walked  each  day. 

26.  A  farmer  rode  in  a  carriage  from  his  home  to  the 
railroad,  and  then  he  rode  on  the  cars  5  times  as  far  as  in 
the  carriage,  and  then  on  a  steamboat  8  times  as  far  as 
in  the  carriage  and  on  the  cars  together,  when  he  had 
traveled  in  all  270  miles.  How  far  did  he  live  from  the 
railroad  ? 


CHAPTER  V. 

MULTIPLICATION. 

EXERCISE  23. 

General  Definition  of  Multiplication. 

34.  The  original  meaning  of  multiplication  in  Arith- 
metic is  that  of  repeated  addition,  and,  with  this  meaning 
in  mind,  we  would  define  multiplication  to  be  the  taking 
of  one  number  as  many  times  as  there  are  units  in  an- 
other. Thus,  3  multiplied  by  5  means  3  +  3  +  3  +  3  +  3, 
and  I  multiplied  by  5  means  f  +  f+f+f +  f .  As  soon, 
however,  as  the  multiplier  is  a  fraction,  it  is  found  that 
this  meaning  of  multiplication  does  not  apply;  for  while 
3  can  be  repeated  5  times,  yet  5  caiuiot  be  repeated  \  a 
time,  nor  can  f  be  repeated  i  a  time.  Now,  although 
the  operation  of  multiplying  |  by  f  cannot  be  looked 
upon  as  repeated  addition,  yet  this  operation  does  occur 
in  Arithmetic,  and  is  called  multiplication.  It  is  plain, 
therefore,  that  the  word  is  used  with  some  other  mean- 
ing than  that  originally  given  it,  which  new  meaning 
may  be  stated  as  follows  :  * 

"  To  multiply  one  nutnber  by  another,  we  do  to  the  first 
what  is  do?ie  to  unity  to  obtain  the  second. ' ' 

Thus,  suppose  we  are  required  to  multiply  3  by  5.  To  make  5  from 
unity,  we  must  take  unity  5  times,  and  hence  to  multiply  3  by  5  we 
must  take  three,  5  times 

Again,  suppose  we  wish  to  multiply  f  by  5.  To  make  5  from  unity, 
we  must  take  unity  5  times,  and  hence  to  multiply  |  by  5  we  must 
take  two-thirds,  5  times. 

*Boset  Algebra  Elementaire,  Charles  Smith's  Elementary  Algebra. 


GENERAL    DEFINITION.  5  I 

Suppose  we  are  required  to  multiply  5  by  f.     To  make  |  from 
unity,  we  must  divide  unity  into  4  equal  parts  and  take  the  result 
3  times.     Hence  to  multiply  5  by  |  we  must  divide  5  into  4  equal 
parts  giving  |  and  take  this  result  3  times  ;  that  is 
5  multiplied  by  |  ig  |X3  or  ^£-. 

Finally,  suppose  we  wish  to  multiply  |  by  i.  To  make  i  from 
unity,  we  must  divide  unity  into  5  equal  parts  giving  I  and  take  this 
result  4  times.     Hence  to  multiply  |  by  |  we  must  divide  |  into  5 

2  2 

equal  parts  giving  , or  -—  and  take  this  result  four  times  ;  that  is 

t)  X  o        10 

-  multiplied  by  -  is  — —  X4  or  —  . 

o  .)       3  X  O  A  O 

1.  What  must  you  do  to  unity  to  produce  8  ?  Explain, 
then,  how  5  is  multiplied  by  8.  Explain  how  %  is  mul- 
tiplied by  8. 

2.  What  must  you  do  to  unity  to  produce  ^?  Explain, 
then,  how  7  is  multiplied  by  \.  Explain  how  f  is  mul- 
tiplied by  \. 

3.  What  must  3'ou  do  to  unity  to  produce  f  ?  Explain, 
then,  how  9  is  multiplied  by  f.  Explain  how  f  is  mul- 
tiplied by  ^. 

Explain,  by  the  general  definition  of  multiplication, 
how  the  product  is  found  in  each  of  the  following  cases: 

4.  11  multiplied  by  12.         13.  6  multiplied  by  6. 

5.  11  multiplied  by  1.2         14.  f  multiplied  by  |. 

6.  3  multiplied  by  |-.  I5-  f  multipHed  by  |. 

7.  i}  multiplied  by  |-.  16.  15  multiplied  by  f. 

8.  I  multiplied  by  3.  17.  1.6  multiplied  by  .25 

9.  f  multiplied  by  f.  18.  12i  multiplied  by  f. 

10.  f  multiplied  by  f .  19.  f  multiplied  by  12^. 

11.  f  multiplied  by -f.  20.   100  multiplied  by  .01 

12.  4  multiplied  by  ^.  21.   .001  multiphed  by  .01 


52  MULTIPLICATION. 

EXERCISE  24. 

Multiplication  of  Monomials. 

35.  If  an  expression  consists  of  but  one  term  it  is 
called  a  Monomial,  and  if  it  consists  of  more  than  one 
term  it  is  called  a  Polynominal.  Thus,  Qab  is  a  mono- 
mial and  oa'^—4:b-\-2  is  a  polynomial. 

If  a  polynomial  consists  of  just  two  terms,  it  is  called 
a  Binomial,  and  if  it  consists  of  just  three  terms,  it  is 
called  a  Trinomial.  Thus,  3<a^— 4^^  is  a  binomial  and 
Q>x~Axy-\-^  is  a  trinomial. 

While  binomials  and  trinomials  are  each  polynomials,  yet  it  is 
usual  to  apply  the  word  polynomial  only  to  expressions  of  more  than 
three  terms. 

30.  It  is  one  of  the  laws  of  multiplication,  discovered 
in  Arithmetic,  that  the  product  will  be  the  same,  no  mat- 
ter in  what  order  the  factors  are  multiplied  together,  or 
as  is  briefly  stated: 

Multiplication  may  be  performed  in  a?jy  order. 

This  is  called  the  Commutative  Law  of  multiplica- 
tion, and  may  be  illustrated  as  follov/s: 

2  X  5  X  7  X  12=12  X  7  X  2  X  5=7  X  2  X  5  X  12,     etc. 
|x5xi=4x|x5=5xix|=4x5x|,     etc. 
And  if  «,   b^  and  c  stand  for  any  juimbers  zv/iatever,   we 

may  say, 

ahc=hca =cab—ach=hac=cba, 

1.  How  many  times  6  is  5  times  twice  6  ?  How  many 
times  a  certain  number  is  5  times  twice  that  number  ? 
How  many  times  ?^  is  5  X  2n  ? 

2.  How  many  times  .^r  is  dx8x} 

3.  How  much  is  8  X  11^? 

4.  How  much  is  25  X  4<^  ? 


EXAMPLES.  53 

5.  How  mucli  is  15  x  12a;y  ? 

6.  How  many  times   2x3  is  2x5x3?     How   many- 
times  ab  is,  axbb}     How  many  times  xy  is  xxSy'^ 

7.  How  much  is  x^  X  31)'-  ? 

8.  How  many  times  2  x  3  is  5  x  2  x  7  X  3  ?     How  many 
times  ab  is  3^  x  8*^ ?     How  many  times  xy  is  Ixx^yl 

9.  How  much  is  Ojt'X  llj'-  ? 

10.  How  many  times  abc  is  9rX  XOab} 

11.  How  many  times  abxy  is  21ay  x  15<^.r? 

37.  From  the  above  work  we  learn  that : 
T/ie  p7-oduct  of  two  monomials  is  found  by  imdtipJying 
the  numerical  coefficients  of  the  monomials  to  obtain  the  nu- 
7nerical  coefficient  of  the  product,  and  by  writing  in  succes- 
sion the  literal  factors  of  the  monomials  for  the  literal  part 
of  the  product. 


EXERCISE  25. 

Examples. 

Multiply 

Multiply 

I. 

^iax  by  9. 

II.   14  by  \ay. 

2. 

8«-U-by  11. 

12.  f\t.y|«.r3. 

3. 

i^/2  by  234. 

13.  mn  by  dy. 

4. 

8^2 jr  by  80. 

14.  xy  by  ab2. 

5. 

lO^V-  by  13. 

15.  ab'^  by  c^y. 

6. 

12^.3  by  |. 

16.  d'^-m  by  hax. 

7. 

25  by  2.r. 

17.   mn'^'  by  9j'2. 

8. 

11  by  \ab. 

x8.  la'^x  by  ^. 

9. 

90by6^;tr2. 

19.   10ayby^2^ 

10. 

15  by  ^77in. 

20.  ^-w  by  9^z/. 

54  MULTIPLICATION. 

Multiply  Multiply 

21.  12a^c  by  4:in.  28.  ^cx^  by  ^Pj/. 

22.  llcfy^  by  So-x"^.  29.  fa'^^w^  by  ^cx. 

23.  Gf?i^p^  by  7d;?2^.  30.  ^d'^n^  by  -f-^c^^n, 

24.  15^-^"*  by  3<^*2^.  31.  -f^VbyyV^^^- 

25.  f^2;rs  by  f^3<5^  32.  i|^^2  by  l^yH"^. 

26.  -fw^^es  by  |^>.  33.  4.6^2^5  by  .Za'^b. 

27.  %b''- y  by  \a^y^.  34.  .5«2;r  by  .6/^^j/. 

EXERCISE  26. 

Law  of  Exponents  in  Multiplication. 

1.  What  is  the  product  of  aaa  and  aa,  written  in  the 
abbreviated  form?  What,  then,  is  the  product  of  «^  and^^  ? 

2.  What  is  the  product  of  ccc  and  cccc,  written  in  the 
abbreviated  form  f  What,  then,  is  the  product  oic^  and  c"^  ? 

3.  In  y^ ,  how  many  times  is  y  used  as  a  factor?  In 
jj/*,  how  many  times  is  y  used  as  a  factor?  In  y^  times 
J*,  how  many  times  is  y  used  as  a  factor?  How,  then, 
would  you  write  the  product  jm^  Xj/*  in  the  simplest  form  ? 

4.  In  tf^,  how  many  times  is  a  used  as  a  factor?  In 
a^ ,  how  many  times  is  a  used  as  a  factor  ?  In  a^  times  a^, 
how  many  times  is  a  used  as  a  factor?  Write  a^  Xa^  in 
the  simplest  form. 

How  is  the  exponent  in  this  simplest  form  obtained 
from  the  exponents  5  and  3  ? 

5.  In  ^^,  how  many  times  is  a  used  as  a  factor?  If  n 
stands  for  a  certain  whole  number,  how  many  times  is  a 
used  as  a  factor  in  a"?  In  a^  times  a'\  how  many  times 
is  a  used  as  a  factor  ? 

How,  then,  would  the  exponent  of  the  product  be 
formed  from  the  exponents  5  and  n  ? 


EXAMPLES.  55 

6.  If  n  stands  for  a  certain  whole  number,  how  many 
times  is  a  used  as  a  factor  in  a"  ?  If  r  stands  for  some 
other  whole  number,  how  many  times  is  a  used  as  a  factor 
in  «''?  In  «"  times  a' ,  how  many  times  is  a  used  as  a 
factor  ? 

How,  then,  would  the  exponent  of  the  product  of  any 
two  powers  of  a  be  found  from  the  exponents  of  the 
factors  ? 

38.  From  the  above  we  learn  the  Law  of  Exponents 
in  Multiplication,  which  is  usually  stated  as  follows  : 

The  product  of  two  powers  of  the  same  number  is  equal  to 
that  7iumber  with  an  exponent  cq^tal  to  the  sum  of  the  ex- 
poneiits  of  the  two  factors. 

39.  In  mathematics  statements  like  the  above  are  often 
expressed  in  the  symbolic  language  of  Algebra,  and  when 
thus  expressed  are  called  Formulas.  The  above  law 
expressed  as  Oi  formula  would  be, 

a"  w  =  a"+^ 

Since  a  is  a72/y  mimber  and  7i  and  r  are  any  whole  7ium- 
bers,  this  algebraic  equation  is  equivalent  to  saying  : 

The  product  of  two  powers  of  the  same  number  is  eqiial  to 
that  number  wi^h  an  exponent  equal  to  the  sum  of  the  ex- 
ponents of  the  tivo  factors. 

EXERCISE  27. 

Examples. 
Multiply  Multiply 

1.  3«2   by  11^3^  5,    5^2^2  by  ^ab. 

2.  bx^  by  Ibx''.  6.  lab^  by  Sa^b'^ 

3.  14^2  by  cb^,  7.   llxy''  by  Sjt^^^ 

4.  14r^2  by  b^.  8.  axy  by  xyz. 


$6  MULTIPLICATION. 

Multiply  Multiply 

g.  adc  by  dcd.  i8.  20<^2^  by  .3^2^. 

10.  8ax^  by  6dx^.  ig.   5w.i:  by  llb'^x^. 

11.  12afy'^  by  10acy\  20.  S^i-^^^  by  20?ix^. 

12.  |<T-.r  by  4<2jv-2.  21.  4??2?i'^  by  pft- 2"^ . 

13.  I^^^j'^  by  4^^jj'*.  22.   2a*_>'3  by  «j'^. 

14.  Sa-x^  by  da^pi-x"^,  23.  2Jy^;r  by  o^^^-j'"^. 

15.  6/>2jt:5  by  Dd^pKr\  24.  33«^  by  IQi^V*. 

16.  Sd^byl-aH\  25.   (;t-4-jiO'  by  4(;r+7)^ 

17.  60xy'  by  .05a-3j.  26.  3(;r-5)2  by  a(ix-5y, 

EXERCISE  28.* 

Multiplication  of  Polynomials  by  Monomials. 

1.  How  much  more  than  200  is  2  times  (100  +  12)? 
How  much  more  than  300  is  3  times  (100+12)  ?  How 
much  more  than  i300  is  5  times  (100+12)?  How  much 
more  than  ?i  hundred  is  71  times  (100  +  12)  ? 

2.  How  much  more  than  200  is  2  times  (100  +  ^)? 
Write,  then,  what  2  x  (100  + (^)  equals. 

3.  How  much  more  than  600  is  6  times  (100  +  r)  ? 
Write,  then,  what  6x(100  +  r)  equals. 

4.  How  much  more  than  2a  is  2  times  (^  +  12)? 
Write,  then,  what  2x(«  +  12)  equals. 

5.  How  much  more  than  4a  is  4  times  (a-}-9)?  Write, 
then,  what  4x((2  +  9)  equals. 

6.  How  much  more  than  /la  is  21  times  («  +  9)  ?  Write, 
then,  what  7tX  («+9)  equals. 

7.  How  much  more  than  91a  is  7i  times  (^  +  15)  ?  Write, 
then,  what  ?i  X  (a +  15)  equals. 


MULTIPLICATION    OF    POLYNOMIALS.  5/ 

8.  How  much  more  than  na  is  n  times  {a-\-b)'^.  Write, 
then,  what  ;^  X  {a-\-b)  equals. 

9.  How  much  less  than  200  is  2  times  (100—12)  ? 
How  much  less  than  300  is  3  times  (100—12)?  How 
much  less  than  500  is  5  times  (100—12)?  How  much 
less  than  n  hundred  is  n  times  (100—12)  ? 

10.  How  much  less  than  200  is  2  times  (100— <^)  ? 
Write,  then,  what  2x(100— /^)  equals. 

11.  How  much  less  than  600  is  6  times  "(100— r)  ? 
Write,  then,  what  G  x  (100— <:)  equals. 

12.  How  much  less  than  2^  is  2  times  (^—12)  ?  Write, 
then,  what  2  x  («  — 12)  equals. 

13.  How  much  less  than  4«  is  4  times  (a— 9)  ?  Write, 
then,  what  4  x  («— 9)  equals. 

14.  How  much  less  than  na  is  n  times  (^—9)  ?  Write, 
then,  what  ;zx(a— 9)  equals. 

15.  How  much  less  than  na  is  n  times  {a—\h)  ?  Write, 
then,  what  ny.  (a— 15)  equals. 

16.  How  much  less  than  na  is  n  times  {a—b)  ?  Write, 
then,  what  ;^  x{a—b)  equals. 

40.  The  principles  we  learn  above  may  be  stated  as 
follows  : 

The  product  of  the  sum  of  tivo  numbers  by  a  third  nu7nber 
equals  the  sum  of  the  products  of  each  of  the  tzco  numbers 
by  the  third  number. 

The  product  of  the  differeiice  of  two  ^lumbers  by  a  third 
number  equals  the  difference  of  the  products  oj  each  of  the 
two  numbers  by  the  tlm^d  7iumber. 

These  principles  may  be  stated  in  algebraic  language, 
by  means  of  the  following  formulas : 
fi{a-\-b) = tia-\-nb, 
n[a—h)=7ia—nb. 


58  MULTIPLICATION. 

We  will  now  proceed  to  the  case  where  the  multipli- 
cand has  more  than  two  terms. 

17.  How  much  more  than  5(a-\-b)  is  5(«  +  <^4-3)  ?  How 
much  more  than  5(«  +  ^)  is  5(a-\-b-\-c)?  Write  an  expres- 
sion equal  to  5(a-{-d-\-c)  without  using  a  parenthesis. 

18.  How  much  less  than  5(«  +  <^)  is  5(aH-<^— 3)  ?  How 
much  less  than  6(a-\-d)  is  b(a-\-d—c)  ?  Write  an  expres- 
sion equal  to  5(a-\-d—c)  without  using  a  parenthesis. 

19.  How  much  more  than  l(a—b)  is  7(a— <^+4)  ?  How 
much  more  than  l{a—b)  \sl{a—b+c)l 

Write  an  expression  equal  to  l(a—b-\-c)  without  using 
a  parenthesis. 

20.  How  much  less  than  7(a—b')  is  7(<2— ^— 4)  ?  How 
much  less  than  7(a—b)  is  7(a—b—c)? 

Write  an  expression  equal  to  7(a—b—c)  without  using 
a  parenthesis. 

21.  How  much  more  than  n(a  +  b)  is  n(ia  +  b-\-c)?  How 
much  more  than  n(a—b)  is  7i{a—b-\-c)? 

Write  what  7i(a-\-b+c)  and  n{a—b-{-c)  equal  without 
using  parentheses. 

22.  How  much  less  than  n(a-\-b)  is  n(a-]  b—c)?  How 
much  less  than  n{a—b)  is  n{a—b—c)t 

Write  what  7i{a-\-b—c)  and  n{a—b—c)  equal  without 
using  parentheses. 

41.  What  we  have  learned  may  be  expressed  by 
formulas  as  follows  : 

n'ya-^-h-^-c) =na-\-nh-\-nCf 

n{a—b-\-c)=n<i—nJb-\-tiCf 

n{  a-\-b — c)  —  na-\-nb — wc, 

Qila —h—c)~  na —nh— nc, 
and,  by  a  continuation  of  the  questions  above  given,  it  is 
easy  to  see  that  we  would  find, 


ARRANGEMENT   OF    WORK.  59 

n{a-{-b—c+d')  =  na-\-nb—nc-\-nd, 
n{a—b-\-c—d')=^na—nb-\-nc—nd, 
n{a—b-\-c-\-d—e)  =  7ia—nb-\-nc-\-7id—7ie, 
and  so  on.     Embodying  all  these  in  a  single  statement, 
we  can  say  : 

The  product  of  a  poly7iomial  by  a  77i07i077iial  is  the  agg7'e- 
^ate  obtai7ied  by  placi7ig  the  i7iultiplier  as  a  factor  i7i  each 
term  of  the  poly7io77iial. 

EXERCISE  29. 

Examples. 

Write  the  product  in  each  of  the  following  examples 
without  using  a  parenthesis  : 

1.  7(2;ir-4y+^).  7.  7^(^-2^+3^-^). 

2.  Za{\g-\-1h-hk).  8.   3«(;i:-4y2  +  2^-8«2). 

3.  1x{\^a—iSb—Zc),  9.  ab{7ir-Yr—7i—^). 

4.  6>'(5?/— 6r-|-4/).  10.  2arm(J'—a-{-r—7n). 

5.  2jf(;r-4ji;2+6).  11.  IxiZ-x-^-^x'^-^x^). 

6.  Aax{Z  +  4:X-S)x'').  12.   day{ll—4y-\-2y''-{-5y^^, 

EXERCISE  30. 

Arrangement  of  Work  in  the  Multiplication  of  a  Poly- 
nomial BY  A  Monomial. 

42.  The  following  illustrations  will  be  sufficient  to 
explain  the  usual  arrangement  of  work  when  the  product 
of  a  polynomial  by  a  monomial  is  sought : 

Suppose  it  is  required  to  multiply  a-{-b—chy  5. 
Multiplicand,         a-\-   b—  c 
Multiplier,  5 

Product,  ba-\-K)b—bc     * 


6o 


MULTIPLICATION. 


Suppose  it  is  required  to  multiply  x—Sx'^-\-2x^  by  bx. 
Multiplicand,         x  —  Sx^ -\-  2x^ 
Multiplier,  6x 

Product,  5;t-2  —  15x-'  +  10;r* 

Suppose  it  is  required  to  multiply  6aj'^  by  2a-j'^  —Say 
+  5.     Since  ^mdtiplication  may  be  pei'foi'mcd  in  any  order ^ 
we  arrange  this  just  as  though  G<rj'-  were  the  multiplier. 
^a^-y"--   Say  +   5 
Ga  y'^ 

Ua'y^-lSa'-y^-^SOay'' 
This  arrangement  of  work    is  analogous   to    the    arrangement    of 
similar  work  in  Arithmetic,  as  the  following  example  will  show  : 

Hundreds.     Tens.     Units. 
Multiplicand,    7/i-^2^-\-'dti  7  2  3 

Multiplier,          3  3 


Product,            21/i-\-ii^-\-9u           21             6             9 
Examples. 

Multiply     x''+2xy+z           Multiply     x''-\-2x-{-S 
by               7                              by              Sx 

• 

2. 

Multiply     2x-by+S 
by              11 

4. 

Multiply     3a^~—4a—Q 
by                4ax 

Multiply 

5.  2x—4y-\-z  by  Sx. 

6.  a^-2ad-d'-  by  aH\ 

7.  ^Vby  4a  +  2ar^-3^'*-l. 

Remember  that  it  is  indifferent  which  expression  is  used  as  the 
multiplier. 

8.  Syx""  by  x^-\-4x'^—dx+6. 

9.  ay^+Sdy'-  —  2fy—12dbyl0ny^. 
10.  Qa'^n  by  S?r-—7a'^-\-47i— 2a. 


EXAMPLES.  6 1 

Multiply 

11.  6ax'^  —  Sa'^x-\-10axhy5ax^. 

12.  I'lx^y^  by  by-\-Q>xy—12x'^y'^. 

13.  x^-Zx'^y-\-Zxy'^-2hYZxy'^. 

14.  Aab  by  'la-d^+oad^—ld"^. 

15.  2«-^-3r^3^i«^3_5by  (3^^2^2^ 

16.  3a/^+4«2^-5^(^+6by  4a*fl?^ 

17.  Ga^-4d''+2ad^-Sc''hy  ^aHc, 

18.  6;>;-^-8:r2^+2.t72  byx2j/3. 

19.  ^— 5^+ic— ifl'by  24«^r^. 

20.  3^-^  +  9^'^r-^2_27by  1^2^. 
Simplify  each  of  the  following  expressions : 

21.  4(4a—7d)  +  7('2a—od). 

4(4^^-7/0  =  16'' -28/';         7(2/r-5^)  =  l4rt-35/^ 
Therefore,     4{U-'id)-\-l{2n-5^)  =  {]6(i-2S/>)-\-{Ua-'65^). 
Removing  parentheses  and  combining  similar  terms, 
4{4a-/0-f-7(2<?-5/0  =  3(k?-63^. 

22.  4(4^— 7^)  — 7(2a— 7^). 

4(4:a-7^>)  =  }Ga-2Sl>;  7(2^-5/0  =  14^-35/^ 
Therefore,  4(4rt-7/0-7(2rt-5<^)  =  (16«-28/0-(14^/-35/0. 
Removing  parentheses,  =1Q<7 —2H^—l4:a-\-33i. 

Combining  similar  terms,  ■=2ii-\-7/k 

23.  5(x+5)-2Gr-4)  +  3(2,r-l). 

24.  S(7a-4d)-4{oa-\-2b)-2(d-?ya'). 

25.  a(a-\-b—c)  —  d(a—d-\-c)-i-c(a4-0). 

26.  7.r(3_r  +  4_y— G)— 4x7jt'— 3j'+14). 

27.  14(3^  +  4^)-6(2«-6^)-4A-0r2-l). 

28.  6a-'{a''-2ax)-4yiy+4xy)-^^a''-ay-). 

29.  «(rt  — ^  +  <:)  +  K^  — <^  +  0  — ^(^  — ^  +  <^)- 

30.  Sad(a-c)-dc(2d-Sa)-\-Qad^-d''(3a-2c), 

31.  3.r(a  +  ^-2j/)-2X«-3^-3A-)-3/5(jt-+2j/)  +  2ay. 


62  MULTIPLICATION. 

32.  Qab{ab'^—Acd)abc. 

Therefore,  Qab{ab^-  ^cd)abc=  %a^b^c{nb^  -  ^cd) 

-^a^b^c-lia^-b^c^d. 

33.  3(16-8;tr-4:r-H2:r-^)6x2. 

34.  3«(2a2+3«2^-3«^2_432-)3^^ 

35.  ab'^iax-bx  -{-Qax^-bbx'^^iaH. 

36.  \xy-(S-xy'^  +dx^y-7y''^Sx\ 

37.  2aH(iax--bx+b'-yb\ 

EXERCISE  31. 

Multiplication  of  Polynomials  by  Polynomials. 

43.  Just  as  it  is  customarj^  to  write  7(10—3  +  5)  in- 
stead of  7x(10— 3-f-5),  so  usually  the  product  of  two 
polynomials  is  indicated  by  writing  it  like 

(14  +  2-9)(10-3  +  5), 
instead  of  using  the  sign  X  ,  as  in  (144-2  —  9)  x  (10—3  +  5). 
So  the  student  must  remember  that  it  is  the  prod?id  ot 
two  polynomials  which  is  called  for  when  they  are  en- 
closed in  parentheses  with  720  sz^?i  between  them. 

1.  If  a-i-b-\-c  multiplied  by  ?i  is  a?i -{- b?i -\- c?7 ,  what  is 
a-{-b-\-c  multiplied  by  («  +  r)  ? 

Write  a(7t-\-r)-{-b(^?t-hr)-\-c(7i  +  r)  without  using  paren- 
theses. 

2.  If  a—b-\-c  multiplied  by  71  is  an  —  b7i-\-c7i,  what  is 
a—b-\-c  multiplied  by  (;z  +  r)  ? 

Write  a{7i-{-r)  —  b{7i-\-r)-\-c{7t-\-7^)  without  using  paren- 
theses. 

3.  If  a-^b—c  multiplied  by  7i  is  a7i-\-bii—C7t,  what  is 
a  +  b—c  multiplied  by  (;^— r+/)  ? 

Write  a(7i  —  r-]-  t')-{-b{7i—7'-\-t)—c{7i—r-\-  t)  without 
using  parentheses. 


EXAMPLES.  63 

4.  Could  results  similar  to  those  above,  be  obtained 
710  matter  what  polynomial  is  used  as  a  multiplier  ? 

44.  As  the  conclusion  from  the  above,  we  have  the 
following  principle : 

The  product  of  two  polynoviials  is  the  aggregate  obtained 
by  placing  one  polynomial  as  a  factor  i7i  each  term  of  the 
other  polynomial. 

EXERCISE  32. 

Examples. 

Find  the  product  in  each  of  the  following  : 

I.   (^-4)  Cr-h9). 

Placing  the  second  polynomial  as  a  factor  in  each  term  of  the  first 
polynomial,  we  have  x(jf-|-9)  — 4(x-|-9). 

Multiplying  by  x  and  4,  we  obtain 

(xs-fy.r)- (4^-1-36). 
Removing  parentheses,  we  get 

,rS-l-9.r-4x-36. 
Uniting  similar  terms,  we  have 

Ar8+5-r-36 
which  is  the  required  product. 

Placing  the  second  polynomial  as  a  factor  in  each  term  of  the  firs! 
polynomial,  we  have 

Multiplying  by  3(^?*,  la  and  9,  we  obtain 

(6^3/;_j8rtV;)4-(4a8^--12r?/^)-a8.-?^^-54/;). 
Removing  parentheses,  we  get 

6«V;-18^?~/^-}-4a2<5— 12(/^-18^^-f-54^. 
Uniting  similar  terms,  we  have 

^a^b—\^a^b—Z^ab^h^b 
which  is  the  required  product. 

3.  (-r+3)(;r+7).  6.   (;t:-3)(^--7). 

4.  (^+3)(-r-7).  7.  (^+5)0r-5). 

5.  (jt:-3)(x  +  7).  8.    (a^+6)(:ir+6). 


64  MULTIPLICATION. 

9.   (x+SXx-4).  20.  (4d-5c)(Sd+4:c).  ' 

10.  (jtr— 5)(jt-— 5).  21.  (Sa—4d)(a—d). 

11.  (^+12)(r-l).  22.  (5.r+l)(7y-2). 

12.  (jc— 12)(;»;— 1).  23.  (^— 5w,)(«H-37;?). 

13.  (x+lo)Cr-lo).  24.  (8a  +  5jr)(7^-4;t:). 

14.  (J^;-^7)(-^--18).  25.  (2^-3Z')(5;r-7:iO. 

15.  (3jt:— 5j/)(3x4-5>').  26.  (#?— ;z)(x+j'). 

16.  (3x— 5jr)(3j»;— 5>').  27.  (a  +  <^)(r— ^). 

17.  (3ji;-57)(5a--3jjO.  28.  (2.r2-4;c+9)Cr-7). 

18.  (2a-2dX2a-d).  29.  (3.r-4-2jr-6)(2.r-3). 

19.  (4;t:+9;/)(x— oj}').  30-  {x'^~xy+y^')(x—y'). 

31.  (3;i:2-2.ry  +  6)/2)(2;r  +  3j/). 

32.  (a^-2ab-hd'-')(a-d). 

33.  (3x2-4;t-  +  7)(ox2-a'--4). 

34.  (^2_|.7_^._5^(^_;^2_3_^_l_7^_ 

35.  (Sa'^-5ad+2d'')(a'--7ad). 

36.  (2jrF— jj/— x)(a-— 3+>')- 

EXERCISE  33. 

Arrangement  of  Work  in  the  Multiplication  of  Two 
Polynomials. 

45.  Let  us  go  through  the  work  of  multiphdng  the 
two  polynomials  7x'^—Qx—^andox'^—bx-{-2  together. 
We  first  place  the  second  polynomial  as  a  factor  in  each 
term  of  the  first  polynomial,,  and  obtain 

7x\Sx''-dx-{-2)-6x<iSx''-5x+2) 

-d(3x''-6x  +  2).  (1) 
Then  multiplying  the  expressions  in  the  parentheses  by 
7x'^,  6x,  and  9  respectively,  we  obtain 

(21x^-^5x^-i-Ux'')-(lSx^-S0x''-\-12x} 

_(27x2-45a^+18).     (2) 


ARRANGEMENT    OF    WORK.  65 

Remov'ing  the  parentheses,  we  have 

-27-^2  H-45;»:-18.  (3) 
Now,  uniting  similar  terms,  we  get 

21x*-r>Sx^-j-17x''-\-SSx-lS  (4) 
which  is  the  required  product. 

This  same  work  can  be  arranged  in  a  convenient  form, 
as  follows  : 

Multiplicand,              3x^—  6x  +  2  (5) 

Multiplier,                   7x^-—  Qx  —  9  (6) 

1st  partial  product,  (21j«;*--3o;t:3  + 14^2)  (7) 

2d  partial  product,         -(IS;*;^— 30;»;2  +  12;ir)  (8) 

3d  partial  product, —  (27.^2  ^45j»;-f  18)  (9) 

where  the  parentheses  are  the  same  as  in  (2)  above,  the 
only  difference  being  that  they  are  arranged  so  as  to  bring 
sbnilar  terms  in  the  same  vertical  column.  Removing  these 
parentheses  and  uniting  the  similar  terms  in  the  same 
column,  we  have 

Multiplicand,  Zx"--  hx  +  2  (10) 

Multiplier,  Ix''-  ^x  -  9  (11) 

1st  partial  product,  2U-*-35j»;3  +  14j»;2  (12) 

2d  partial  product,  —X'^x^ -^Z^x"- —  Vlx  (13) 

3d  partial  product,  • -27jt:2-f  45;t;~18        (14) 

Product,  21  j«;*  -  53;r3  +  17;t:2  ^  33;r- 18         (15) 

which  is  the  usual  arrangement  of  work  in  the  multipli- 
cation of  two  polynomials. 

46.  The  expressions  (12),  (13),  and  (14),  which  we 
have  called  partial  products,  were  obtained,  as  we  have 
just  stated,  in  the  following  manner: 

First.  The  expression  ?>x'^  —  hx-{-2  ivas  placed  as  a  factor 
in  each  term  of  the  expression  7;»:2— 6.v— 9. 


66  MULTIPLICATION. 

Second.  The  7miltiplications  by  Ix"^,  6x,  a?id  d  were  per- 
formed. 

Third.    The  parentheses  were  removed. 

Hence  it  follows  that  when  the  multiplicand  is  placed 
as  a  factor  in  the  additive  term  Tjt^,  the  result  is  the 
partial  product  (12),  which  has  its  terms 

additive,          subtractive,         additive, 
the  sa7ne  as  the  multiplicand.   But  when  the  multiplicand 
is  placed  as  a  factor  in  the  subtractive  terms  —%x  and 
— 9,  the  results  are  the  partial  products  (13)  and  (14), 
which  have  their  terms 

subtractive,          additive,         subtractive, 
which  are  just  the  opposite  of  those  of  the  multiplicand. 

47.  From  this  reasoning  we  can  readily  formulate  the 
following  method  for  finding  the  product  of  any  two 
given  polynomials  : 

Neglect  the  sig7is  of  the  terms,  and  miiltiply,  in  succession, 
the  multiplicand  by  each  term  of  the  fnultiplier,  to  obtain  the 
successive  partial  products. 

Give' the  terms  of  each  partial  product,  the  same  sights 
as  those  of  the  multiplicand  when  an  additive  term  of  the 
multiplier  is  used,  but  give  the  terms  of  each  partial  pro- 
duct just  the  opposite  signs  to  those  of  the  multiplicand  whefi 
a  subtractive  term  of  the  multiplier  is  used. 

Add  the  partial  products  thus  formed,  and  the  result  is 
the  required  product. 

Thus,  suppose  (x^-\-'^x—^{x~  —  ^-)X-\-1)  is  required, 
x2+3x  -  4 
x^—hx  +  2 

—hx^  —  \hx^^l{)x 

2x8+   (;^_8 


;c4— 2x3  — 17x3-f-2G^— 8 


ARRANGEMENT    OF    WORK.  6/ 

The  above  way  of  obtaining  the  signs  of  the  terms  in  the  partial 
products  keeps  the  ''reasons  why  prominently  before  the  student,  but 
in  practice  the  signs  are  more  often  determined  in  another  manner, 
which  we  proceed  to  explain. 

48.  In  the  above  we  have  seen  that  the  signs  in  the 
partial  products  will  be  the  same  as  those  of  the  multi- 
plicand if  the  term  used  in  the  multiplier  is  additive. 
Hence  we  have  the  two  following  cases  : 

If  a  term  in  multiplicand  is  additive,  i.  e. ,  has  sign  -f 
and  term  used  in  multiplier  is  additive,  i.  e.,  has  sign  + 
the  resulting  term  in  product  is  additive,  i.  e. ,  has  sign  + 

If  a  term  in  multiplicand  is  subtradive,  i.  e.,  has  sign  — 
and  term  used  in  multiplier  is  a^^zVzV^,  i.  e.,  has  sign  -f 
the  resulting  term  in  product  is  subtradive,  i.  e.,  has  sign  — 

We  have  also  learned  that  the  signs  in  the  partial 
products  will  be  the  opposite  to  those  of  the  multiplicand 
if  the  term  used  in  the  multiplier  is  subtractive.  Hence 
we  have  the  two  following  cases  : 

If  a  term  in  multiplicand  is  ^^^///z'<?,  /.  e,,  has  sign  + 
and  term  used  in  m.\.\\X.\\A\Qr  is  subtradive,  i.  e.,  has  sign  — 
the  resulting  term  in  product  is  siibtradive,  i.  e.,  has  sign  — 

If  a  term  in  multiplicand  is  subtradive,  i.  e. ,  has  sign  — 
and  term  used  in  multiplier  is  5?/<^/;'^<://z'^,  i.  e.,  has  sign  — 
the  resulting  term  in  product  is  additive,      i.  c. ,  has  sign  + 

From  this  we  see  that  when  the  terms  in  multiplicand 
and  multiplier  are  bot/i  additive  or  bot/i  subtractive,  the  re- 
sulting term  in  the  product  is  additive  ;  but  when  the 
terms  in  multiplicand  and  multiplier  are  o?ie  additive  and 
07ie  sicbtradive,  the  resulting  term  in  the  product  is  sub- 
tractive.  From  this,  results  the  statement  known  as  the 
Rule  of  Signs : 

I?i  multiplication,  like  sig7is  give  plus  and  ujilike  signs 
^ive  7iii7ius. 


6S  MULTIPLICATION. 

49.  Now  the  method  given  in  the  previous  article 
may  be  stated  as  follows  : 

To  Multiply  two  Polynomials  together,  7ieglect  the 
signs  of  the  terms  and  multiply,  in  succession,  the  multipli- 
cand by  each  term  of  the  mtdtiplier,  to  obtain  the  successive 
partial  products. 

Determine  the  sig7i  of  each  term  of  the  partial  products  by 
the  Rule  of  Signs. 

Add  the  partial  products  thus  formed  and  the  result  is  the 
required  product. 

60.  The  following  examples  will  tend  to  show  the 
advantage  of  arranging  the  work  of  multiplication  in  the 
way  explained: 

x'^-\-  X  y-\-  j)/2 
X  —      y 
x^-\-x'^y-\-xy^ 
— x'^y — xy^  —y^ 


X  +    y 

a  -   b 

X  +    y 

a  +   b 

x'^-h  xy 

a'^—ab 

xy-hy' 

ab-b'' 

x^  +  2xy+y'^ 

«2               _^2 

x^  —y^ 

♦*The  student  should  observe  that* with  the  view  of  readily  bring- 
ing similar  terms  of  the  product  into  the  same  column,  the  terms  of 
the  multiplicand  and  multiplier  are  arranged  in  a  certain  order.  We  fix 
on  some  letter  which  occurs  in  many  of  the  terms,  and  arrange  the 
terms  according  to  the  powers  of  that  letter.  Thus,  taking  the  last  ex- 
ample, we  fix  on  the  letter  x  ;  we  put  first  in  the  multiplicand  the  term 
x^,  which  contains  the  highest  power  of  x,  namely  the  second  power; 
next  we  put  the  term  xy  which  contains  the  next  power  of  x,  namely 
the  first;  and  last  we  put  the  term_j/*  which  does  not  contain  x  at  all. 
The  multiplicand  is  then  said  to  be  arranged  according  to  descending 
powers  of  x.     We  arrange  the  multiplier  in  the  same  way. 

We  might  also  have  arranged  both  multiplicand  and  multiplier  in 
reverse  order,  in  which  case  they  would  be  arranged  according  to 
ascending  powers  of  x.  It  is  of  no  consequence  which  order  we 
adopt,  but  we  should  take  the  same  order  for  the  multiplicand  and  the 
multiplier,"  —  Todhunter'' s  Algebra  for  Beginners. 


EXAMPLES.  69 

EXERCISE  34. 

Examples. 

Multiply  the  following  expressions: 

1.  A--13by  ;»;-14.  8.  5;i;+6  by  2jr— 3. 

2.  x+19by;r— 20.  9.  3a4-7  by  5a  +  9. 

3.  2jir— 3  by  ;»r+8.  10.  7a  +  4  by  ija^ -{-Sa. 

4.  2a-+3  by  x-8.  ii.  2xj+y-  by  A-2-2jt:j. 

5.  ;r-5by2A-~l.  12.  Sa'^-iahy  2a^ +Qa^-, 

6.  2;ir-5by    ^-1.  13.  3x2+2^2  by  3jr 2- 2^. 

7.  o;»r— 5  by  2;t:+8.  14.  ;«;2— ;9/-f^2  ^y  ^^^^ 

15.  5a^ -\-a  +  S  by  ba-{-Q. 

16.  5  +  2«  +  8^2  by  6  +  8«  +  3«^ 

17.  2«  +  9a2+3a3  by  8  +  4«  +  5«^ 

18.  3— 2;r+.r2  by  5—x. 

19.  x'^—2ax-^5a'^  by  x—Sa. 

20.  7-4;r-3jr2+5;i:3  by2  +  5;r. 

21.  2?^2 — 72^2; — 3z'2  by  3u — 2v. 

22.  5a^—Sax—6x'^  by  oa—5x. 

23.  l-2jt:+3x2  by  24-3;r-4;i:2. 

24.  2;tr2— ;i;+3by  2.r2— 2:r— 4. 

25.  3«2-2a;i:4-7;r2  by  3a2  4-2«A'-7;t-2. 

26.  7«'''  +  2«;»r— 4j»:2  by  7a2— 2«;j;— 4;t:2. 

27.  5a^—2a'^x+ax^  by  2a2— a;»;+2;*:2. 

28.  2;r2— 3  +  5;tr''^  by  6;»:— 8  +  4;t:^ 

Arranging  according  to  the  descending  powers  of  x,  we  obtain 

ix^-j-Gx  -8 

20x8+8a;5  —  12a;« 

+30a;4H-12x»  -18a; 

—-iOx^-l^x^  +24 

20^»+8a;6-|-30a;*-40a:3-T6x«  -18a--(-24 


70  MULTIPLICATION. 

29.  7—4:X-i-Sx^hy5x'^—x—4:. 

30.  7x-5-hx^-  by  7-Sx+x^. 

31.  ^^3^3a3^-2a2^2  by  2a'^-ad-5d^ 

32.  x^  —  2x+l  by  x^— 3jr+2. 

33.  .r2— 5^jr— 2^2  by  ;t-2  4-2«ji'+3a2. 

34.  7-r2+jj/2_3ji:j/  by  2j»:''^ +J-Jr. 

35.  2j»;-^-4jc2_4^._1  by  2x^-4x^-ix-l, 

36.  5X— 7j«;2-}-jt:3-f-l  by  l-\-2x^—4:X. 

37.  n—4x-{-x-  by  4x2  — 24  +  5jtr+3jir. 

38.  4jt:2-3,r)/-j/2  by  3jr-2_y. 

39.  «2_^^_j_^^_^^2    by   ^-j-^-l-.^:^ 

40.  x''--\-y'^—xy+x-\-y—lhyx+y—\, 

41.  x+2j— 3^  by  X— 2j/+32'. 

42.  a2_|.32_^^2_^^_^^_^^by  «  +  ^+<r. 

43.  «^2_2^^-|-^2^.^-2    by  a2+2^^+^2_.^-2^ 

44.  3«2_^2^  +  2a3  +  l  +  a*  by  a2-2«  +  l. 
45-  -^^  +y^-h^^  -{-xy—yz+xz  by  x— j/+^. 

46.  ix2_2^-+3  by|a-+i 

47.  f^+ 2^  by  |a-|^. 

48.  |x-f  by  ix+|. 
49-  i^—iy  by  ^x+y. 

50.  2ix'-3ibylix+fV 

51.  1^2  4.^^  +  ^byi^-i       , 

52.  |«2^|ijby«-|. 

53.  f^2_|^^^.8^by|«2^.|^, 

54.  ix'^-ix-i  by  |;^2^2^_8. 

55.  ix^—%xy+y^  by  fx+^y, 

56.  |;«-|r+f/>  by  i;;^  +  |-r-6A 

57.  i^-#+J^by|«4-f^-i^. 


EQUATIONS   INVOLVING    MULTIPLICATION.         7 1 

EXERCISE  35. 

Equations  Involving  Multiplication. 

Solve  the  following  equations  : 

1.  9(23-5y)=8(5>/-6). 

Performing  the  multiplications  by  9  and  8,  we  have 
207— 4oy=40>'— 48. 
Transposing  the  terms  —48  and  — 45j',  we  get 
207+48r=40>'+45y. 
Uniting  similar  terms,  255=^85v; 

whence,  y=^Z. 

2.  6(jt:-5)  +  2A'=8A'-2(jt-+10). 

Performing  the  multiplications  by  6  and  2,  we  have 
(Ca;-30)+2a:=:8x-(2aH-20). 
Removing  parentheses, 

()X— 30-|-2a:=  8a;— 2x— 20. 
Transposing  the  terms  —30,  8a:,  and  —2x,  we  obtain 

6a; -[-2x- 8-r-f2.r.- 30-20. 
Uniting  similar  terms,  2x=10; 

whence,  x=b. 

3.  2(;ir-l)  =  G.  17.   loCr-3)-17=103. 

4.  4(;»;+5)=36.  18.  17(17~;t:)4-17=51. 

5.  7(j^/-3)=14.  19.  6(37-.;»r)  +  23=113. 

6.  19(j»r-7)=57.  20.  5(2,r+7)-8=57. 

7.  13(12-^)=2G.  21.  9(8^-5) +  13=  112. 

8.  17(13-.r)=136.  22.  5(.r-l)=9A'-25. 

9.  5(35-jt:)=105.  23.  5;»;+(7-2x)=ll. 

10.  7(135-^)=85.  24.  8jtr-(3+5;r)=9. 

11.  8(2:r+5)=lo2.  25.  8(5-^)=3(jj/-5). 

12.  13(7r-61)=26.  26.  10(;t+2)=ll^+17. 

13.  4(15-2.r)  =  20.  27.  4(5jt:-3)=104-9;r. 

14.  15(15-4;r)=45.  28.  8(10-;r)  =  5(.;r+3). 

15.  3(;i:-5)  +  8=17.  29.  7(15-3jr)  =  GCr+4). 

16.  3(j«:-3)  +  5=23.  :o.  8(9-2^0  =5(3;»;+2). 


72  MULTIPLICATION. 

31.  9(13-;r)-4;i;==5(21~2;r)-f9;r. 

32.  199  +  15;»;-(4jr-5)=17(x+17)-13-22. 

33.  8(3jt:-2)-7^-5(12--3jt:)  +  28=8(3x4-2)-32. 

34.  118^-13(54-ll^)  +  15(3;»;-3)=.18;»;--7(^--5)-119 

35.  7(3jr-6)+5(x-3)  +  4(17-x)=44. 

36.  2CW-x)  +  S{5x-i)=:12(S+x)-2(12-x), 

37.  3(3^-2)~5(18-5;^)  =  13(8;r-12)4-5(9;»;~ll). 


CHAPTER  VI. 
DIVISION. 

EXERCISE  36. 

Division  of  Monomials  by  Monomials. 

61.  We  may  define  Division  as  the  process  of  undoing 
multiplication.  In  multiplication  two  factors  are  given 
to  determine  their  product,  while  in  division  the  product 
and  one  of  the  factors  are  given  to  determine  the  other 
factor.  Thus,  to  divide  12  by  3,  we  must  determine  the 
factor  which,  when  multiplied  by  the  given  factor  3,  will 
produce  12. 

In  divison,  the  factor  given  is  called  the  Divisor;  the 
product  given  is  called  the  Dividend;  and  the  factor  to 
be  determined  is  called  the  Quotient. 

1.  What  must  3<z  be  multiplied  by  to  produce  6«? 
What,  then,  is  6a^3«? 

2.  What  must  Za  be  multiplied  by  to  produce  6<?^? 
What,  then,  is  ^ab-^?>a  ? 

3.  What  must  hb  be  multiplied  by  to  produce  Z^abc^  ? 
What,  then,  is  Z^abc"- ^hb'> 

4.  What  must  «^  be  multiplied  by  to  produce  «^  ? 
What,  then,  is  a^^a^'> 

5.  What  must  €>a^x^  be  multiplied  by  to  produce 
r^a^'x'^  ?     What,  then,  is  X'^a'^ x'^ -^Za'' x^  ? 

52-  Since  the  dividend  equals  the  product  of  the 
divisor  and  quotient,  it  follows  that  the  quotient  of  one 
monomial  by  another  monomial  is  found  by  removing  from 
the  dividefid  all  the  factors  which  ocair  in  the  divisor. 


74  DIVISION. 

Thus  ^abc-^2ab—A:C,  because  2abx4:C=^8abc,  and  the 
quotient  Acis  found  by  removing  the  factors  2,  a,  and  b 
from  the  dividend.  If  we  wish  to  divide  2ab  by  xy,  the 
factors  X  and_y  do  not  occur  in  the  dividend,  and  conse- 
quently cannot  be  removed  from  it,  so  we  can  merely  m- 

,      ,.    .  .        ,         2ab 

dicate  the  division  thus: . 

xy 

53.  Since  the  product  of  two  powers  of  the  same 
number  is  found  by  adding  the  exponents,  it  follows  that 
the  quotient  of  any  power  of  a  number  divided  by  a  lesser 
power  of  the  same  number,  is  equal  to  that  nuitiber  with  an 
exponent  equal  to  the  exponent  of  the  dividend  minus  the 
exponent  of  the  divisor.  This  is  called  the  Law  of  Ex- 
ponents in  Division. 

Thus,  a^-r-a^=«^  because  «^Xa^=^^.  Also  2\a^bx^ 
'^Sabx''  =  ^a''x^  because  ^abx"- X?ja''x''  =  Ua^bx\ 

54.  The  above  principle  may  be  stated  in  algebraic 
language  by  means  of  2i  formula,  for  if  we  let  a  stand  for 
any  number  whatever,  and  n  and  r  stand  for  a7iy  whole 
numbers  whatever  such  that  71  is  greater  than  r,  we  may 
say, 

because  a"~''  x  a'  —  a". 

EXERCISE  37. 

Examples. 
Divide  Divide 

1.  6^  by  3.  6.   14r^by  d. 

2.  8b  by  4.  7.   17m?i  by  m. 

3.  1577in  by  5.  8.  2Sxy  by  x. 

4.  49^^  by  7.  9.   2o77ix  by  ox. 

5.  12ab  by  a.  10.   dAa^y  by  6y. 


EXAMPLES. 


75 


Divide 

11.  IS^^jt:^  by  63. 

12.  28cy^  by  4c. 

is.   lOSm^c^  by  9;«^ 

14.  I(y5dm''  by  15/^. 

15.  76dx^-  by  IM 

16.  51w2jj/  by  17;«2. 

17.  6Smn^  by  9;«. 

18.  145;;^;z-j/  by  29;/?. 

19.  5cy  by  r>'. 

20.  4Sm^7t  by  ;;^2;^. 

21.  Ilx^y  hy  12x'\ 

22.  17;t:8j'  by  17;»;». 

23.  53;»:*  by  5;t*. 

24.  Ilx^y-  by  Tljj''-'. 

25.  2;«^2'  by  2m^z. 

26.  475^2  by  2^2^ 

27.  17a2<^  by  a^^. 

28.  423^2  by  be'',     • 

29.  2«'m2  by  ;;2. 

30.  5a ^^"^  by  a^. 

31.  Sm^fi^  by  «^. 

32.  15^/^"^  by  3«(^. 

33.  5/^^jr*  by  ;i:^. 

34.  18^2^-5  by  3j'^ 

35.  Sm^7i^  by  /z*'^. 

36.  9m*jy^  by  Sm'^y^, 

37.  ^^/^^  by  «=^3"^ 

38.  Ibv^x  by  5z;2;,;. 

39.  «2^^3  ]3y  ^^2 

40.  ^V;t*  by  d^x'^. 


Divide 

41.  Id^x'^  by  ^3 1-2. 

42.  aH^  hy  ab. 

43.  fn^7iy^  by  ;r^2^ 

44.  21z;*J»;2_y5   |)y  72;3;^2^ 

45.  6j»;7_>/9  by  Zxy^. 

46.  8^^ji:y  bj2r2j3. 

47.  hXb^x^z  by  l?^^;^;^ 

48.  5a;i:^  by  hax^ . 

49.  15^^*  by  5fl^3. 

50.  225w^j^'^  by  lowy*. 

51.  VlXx-'y^  by  ll;i:-^j2 

52.  lOOS^^VoT^  by  18/^^^. 

53.  306^«*^iJ/>  by  18w/. 

54.  10^2^  by  5  ab. 

55.  140;t:3j/5  by  Zhx^-y. 

56.  108/i2j^/7  by  9/;>'«. 

57.  702^5^*  by  18^/. 

58.  XOmx^y'^  by  SS^x'^y^, 

59.  f^2^4  by  5  xy, 

60.  fm^y^  by  ^m*y^. 

61.  fa7^2  by  yV^^. 

62.  T-V/^2^  by  i/>. 

63.  ^d-^c^e  by  yV^r. 

64.  i/'V^^  by  f/)=^^r. 

65.  ^  kr-  by  f  /&/. 

66.  Ic'^x^  by  Icx"^. 

67.  ia2^3  by  -^\ab^. 

68.  11^4^3  by  -i-l^/z^ji:. 

69.  \ixy^  by  Ji;r>/^ 

70.  .o^h^  by  .4/i*. 


'J^  DIVISION. 

EXERCISE  38. 

Division  of  Polynomials  by  Monomials. 

55.  Since  the  product  of  a  polynomial  by  a  monomial 
is  found  by  placing  the  monomial  as  a  factor  in  each  term 
of  the  polynomial,  it  follows  that  a  poly^iomial  is  divided 
by  a  monomial  by  removing  from  each  term  of  the  polynomial 
all  the  factors  which  occur  in  the  divisor. 

Thus.  (G«-9^+15r)-f-3  =  2^-3^+5^,>r  (2^-3^  +  5^) 
X  3=6«— 9^+15r,  and  the  quotient  is  found  by  removing 
the  factor  3  from  each  term  of  the  dividend. 

Also,  (9^S;r-15^2^2_^12a:r3)-f-3«;t:=3a2_5^^ 4.4^2^ 

because  (3«^— 5ajt:+4jf2)  y.Zax=^^a^ x—\ha'^ x'^ -{-Vlax^ y 
and  the  quotient  is  found  by  removing  the  factors  3,  «, 
and  X  from  each  term  of  the  dividend. 

If  we  wash  to  divide  IQa—dbx-^-Scx"^  by  4x,  the  factors 
4  and  x  do  not  occur  in  some  of  the  terms  and  conse- 
quently cannot  be  removed  from  them,  so  we  can  merely 
indicate  the  division  in  such  cases,  thus  : 

(^iQa-ddx  +  Scx')-r-ix=:^~-~  +  ^-. 

56.  The  Arrangement  of  the  W^ork  when  a  poly- 
nomial is  divided  by  a  monomial  is  as  follows  : 

Let  it  be  required  to  divide  21?n'^x^  —  S5am-\-7h'^m^ 
by  7;«. 

Divisor,     7m  I  21m'^x^  —  S5a7n  +  7h-m'^     Dividend, 
Sm  x^ —  ba     -f  h'^m       Quotient. 
Let  it  be  required  to  divide  l(jx^  ^2Ax^ -20x''  by  Ax^. 
Divisor,  "  Ax^  |  16;t:^  +  24:t-'^— 20^"     Dividend, 
4x'^-h  6x  —  5         Quotient. 
Let  it  be  required  to  divide  5x'*'—7x^y-j-4x^y'^  by  5x^. 
Divisor,     5x^  I  5x'^—7x^y-^4x'^y'^     Dividend, 


r'-i  — I 


Ix  y-h^y'-         Quotient. 


DIVISION    OF    POLYNOMIALS.  TJ 

EXERCISE  39. 

Examples. 
Divide 

1.  15«2_9a-5  +  18^»  by  3«2. 

2.  UH-l^a'^b^-^-X^aH^  by  ^a^b, 

4.  6aH'^-Soa^b^c^+20adc'^  by  oad. 

5.  12a''x*y- -24ax^y* -18x''j^+6xy  by  6;rj. 

6.  Sx^y--12x--16x  by  4ji:. 

7.  3a3^2-6a2^*4-12«^6^by  3^/^^ 

8.  Sx^--{-l(Dax-{-6a\rhy4x. 

9.  2«2^"-3«<^3_|_4^3^_^4  by  3a<^^ 

10.  a^x^y—Sa'^dx^j+Sab'^xy^  —  d^xy^  hy  abxy. 

11.  3A'*^+5jr3;j/-6;«:V^— -^y +  4y5  by  2x''y''. 

12.  35:i:^H-15;«:2^--;r>/2--j3  by  5^2. 

14.  5.r^j/— 25;t:2y2 _}_ \0xy^—8  by  5;rj/. 

15.  4;>Y^r2+J^/2*r^— ^w^r^  by  f/;«^r. 

16.  ZQx^y^-JrlSx^y^-Ux^y^  hy  ^x^y^, 

17.  .6m'^x^—f7n'^x'^-{-.Somx^  by  fwx^. 

18.  ^^y^  —  '^x^y—zz^yhy\xy. 

EXERCISE  40. 

Division  of  Polynomials  by  Polynomials. 

67.  Suppose  we  wish  to  divide  x'^-\-bx-\-Q  by  .a:+3. 
The  dividend  x'^-\-bx-\-(S  is  the  product  of  the  divisor 
x-\-Z  and  another  factor  (the  quotient),  which  we  wish 
to  find.  Now,  we  can  tmdo  this  multiplication  if  we  can 
write  x'^-\-bx+Q  so  that  {x-\-Z)  will  be  a  factor  in  each  of 


7^  DIVISION. 

its  terms,  for,  by  the  previous  exercise,  an  expression  is 
divided  by  (:r+3)  if  we  remove  (^+3)  as  a  factor  from 
each  term.     But  we  can  say, 

and,  by  using  parentheses, 

=^(^+3)  +  2(:r+3). 
Then,  by  removing  the  factor  (x-\-Z)  from  each  term,  we 
obtain 

(jr2+5;i:+6)-^(;r+3)  =  *r+2, 
which  is  the  required  quotient. 

As  another  example,  let  it  be  proposed  to  divide 
j»;2  +  14;»;-f  45  by  x-\-^.  Now  this  can  be  done  if  we  can 
write  x'^-]-14iX-\-Ab  so  that  (;r+9)  will  be  a  factor  in  each 
of  its  terms;  for  an  expression  is  divided  by  (jr  +  9),  if  we 
remove  (:r4-9)  as  a  factor  from  each  term.  But  we 
know 

x-  +  14;r+45=x2-i-9j»;+5^+45 
and,  by  using  parentheses, 

=.:r(j»;+9)  +  5(x  +  9). 
Then,  by  removing  the  factor  (;r+9)  from  each  term,  we 
obtain 

(jt:2  +  14;r+45)^(jf+9)=j»;+5, 
which  is  the  required  quotient. 

Again,  suppose  it  required  to  divide  x'^-\-5x-\-4  by 
jt:+4.  This  can  be  done  if  we  can  write  x''-\-5x-\-x,  so 
that  (x-\-4)  is  a  factor  in  each  of  its  terms  ;  for  an  expres- 
sion is  divided  by  (x-j-4),  if  we  remove  (x-i-4)  as  a  factor 
from  each  term.     But  we  know 

x'^-\-5x-{-4=x^-\-4x-\-x-i-4: 
and  by  using  parentheses, 

=x(x+4)  +  (x+i). 
Then,  by  removing  the  factor  (.^+4)  from  each  term,  we 
obtain  (x'^-j-ox-}-4)-^(x-^4')=x-\-l, 

which  is  the  required  quotient. 


EXAMPLES.  79 

It  is  noticed  in  each  of  the  above  cases  that  our  process 
consists  in  breaking  the  given  dividend  into  parts,  so 
that  the  divisor  is  a  factor  of  each  part,  and  then  remov- 
ing that  factor  from  each  term. 

58.  We  may  formally  state  the  above  process  as 
follows  : 

One  polynomial  is  divided  by  a  secona  polynomial,  if  the 
first  polynoinial  be  written  so  that  the  second  polynomial  is  a 
factor  in  each  term  of  the  first,  and  this  factor  removed. 

If  it  is  impossible  to  write  a  polynomial  in  this  man- 
ner, then  that  polynomial  is  not  exactly  divisible  by  the 
proposed  divisor,  and  the  division  must  be  merely  in- 
dicated. Thus,  (;»:-^4-6jir+2)H-(;r+3)  would  be  worked 
as  follows  : 

Using  parentheses  =^(ji;+3)  +  (oji--|-2). 

Therefore, 

EXERCISE  41. 

Examples. 

Divide  Divide 

1.  :i:2+3;r4-2by  ^+2.  g.  jt^+Qjr-fM  by  .r-f7. 

2.  x'^^-^x-k-^hy  x-\-Z.  10.  .;i:2-}-8^-fl5by  j*r-f3. 

3.  x'^-'^^x^hhy  x-\-h.  ii.  x^-\-\\x^1'^hy  x-k-^. 

4.  .r2  +  7.r+10by  .r-f-2.  I2.  .^'M 'J-^-f  18  by  .^+6. 

5.  x'^-\-hx-\-^\yy  x^Z.^  13.  :r2  +  10.r-r24  by  .;»;4-4. 

6.  x'^-^-'dx^l^hy  x^^.  14.  A'2_^10.r+21by.r-t-7. 

7.  ;i:2-f6;ir+8by  ^4-2.  15.  .r^ +  12;<;-f-35  by  .r-fS. 
S.';i:2  +  8;t--f  12  by  ;t:-f6.  16.   x'^  ^-A^x-Yoby  x-\-\. 


8o  DIVISION. 

Divide 

17.  x'^-\-4:X—45  by  x—5: 

Using  parentheses  =x(x—ri)-{-9{x—5). 

Removing  the  factor  (x— 5)  from  each  term 

{x^-\-U-4o)^{x-\-5)=x-\-9, 
the  required  quotient. 

18.  j»;2-f3jtr— lOby  :r— 2.       21.   ji:^— 4j»;— 21  b}^  Jt:— 7. 

19.  x^-\-Sx—4hyx—l.        22.  x'^ +2x—S6  by  x—5. 

20.  Ji;2-f3;»;-18by  ;«;-3.       23.  jr2-4.r-12  by  ;i:-6. 

24.   Divide  ;i:^ — 4;»;— 45  by  .r-f-5. 

x^-4x-45=x^-\-^x-9x-45. 
Using  parentheses  =x[x-{-5)  —  9{x-\-5).  Art.  23. 

Removing  the  factor  {x-\-5)  from  each  term 

(.r8— 4a-— 45)h-(a-4-5)  =  x— 9, 
the  required  quotient. 

25.  ;i;2-3jt:~10by  .:r  +  2.       28.   ;r2-4.r-21  by  ;t:+3. 

26.  .;r2— 3;*;— 4by  .r+1.         29.  j»;2  +  3jtr— 10  by  ;t4-5. 

27.  .;i;2-f3:r-18by  .r+6.       30.   ^•-+2j«;-35  by  ;i:+7. 

31.   Divide  .;>;2  —  14.a; -1-45  by  jir— 5. 
x^-Ux-\-i5=x»-5x-9x-\-45. 
Using  parentheses  =x{x—^)-9{x—5).  Art.  23. 

Removing  the  factor  {x—5)  from  each  term 

{x^-Ux-\-4:5)^{x-5)=x-9 
the  required  quotient. 

32.  a^-7a  +  10   by«-2.   39.  Sa^- +  19a-\-20  by  a-\-5, 

33.  a^  —  5a-{-4:by  a—1.  40.  4:a'^-\-Ua  +  6   by  4a -f  2. 

34.  a'^-da-j-18   bya-3.   41.  4a''-\-2Sa-\-15by  4a-{-S. 

35.  a^-lOa-i-21  bya-7.     42.  3^^ ^10^-^3  by  a-h3. 

36.  a''—7a  +  10   byd;-5.   43.  5tf2  +  lla  +  2  by  «-f-2. 

37.  ^2-10^  +  24  by«-6.  44.  2^^  +  11^  +  5  by  2«4-l. 

38.  2a2-hl0«  +  12  by«  +  3.  45.  3a2  +  34«-i-llby3a-I-l. 

46.  4^2  +  23^4-15  by  4^-1-3.        ♦ 

47.  24a-—66a  +  21  by  8a— S. 


ARRANGEMENT   OF    WORK.  8 1 

EXERCISE  42. 

Arrangement  of  Work  in  the  Division  of  Polynomials  by 
Polynomials. 

59.  Since  division  is  the  process  of  undoing  multipli- 
cation, we  will  exhibit  in  connection  with  each  other  the 
arrangement  of  work  in  the  two  operations. 

multiplication,  or  the  direct  operation. 
Multiplicand,  x  -\-  4 

Multiplier,  .^-  4-  ^ 

1st  partial  product,       x'^-h  4;r 

2d  partial  product,        9^+36 

Product,  x'^-i-lSx+Se 

division,  or  the  inverse  operation. 

Divisor.  Dividend.  Quotient. 

x+A^  x''-j-rSx-hS6  ix+d 
1st  partial  dividend,      .r^-f  4j«;      _ 
1st  remainder,  9jt--f36 

2d  partial  dividend,  9jc+3'6 

2d  remainder,  0 

In  the  work  in  division  the  process  is  as  follows  :  We 
first  arrange  dividend  and  divisor  thus, 

x+A)  x''-\-nx+S6  ( 
We  next  divide  x^,  the  first  term  of  the  dividend,  by  x, 
the  first  term  of  the  divisor,  which  gives  x  as  the  quotient. 
We  now  multiply  the  w/io/e  divisor  by  x  and  put  the 
product,  x'^-\-Ax,  under  the  dividend.     We  then  have 
x^^:)  jr2-f  13.^+36  {x 
x^-\-  Ax 
by  subtraction,  9.;i;-f36 

Then  divide  9jr,  the  first  term  of  this  remainder,  by  x, 
the  first  term  of  the  divisor,  which  gives  9  as  the  quotient. 
Now  multiply  the  whole  divisor  by  9  and  put  the  product, 
9a;-f  36,  under  the  last  remainder.     We  then  have 

6 


82  DIVISION. 

x+4.  )  x^'  +  lSx+SQ  (  x-{-d 
x^-\-  Ax 

9jt:-f36 
by  subtraction,  0 

whence  the  quotient  is  x-\-^. 

B}^  comparing  the  above  work  in  division  with  that  of 
multipHcation,  the  student  will  observe  that  the  partial 
products  which,  when  combined,  constitute  the  final 
product  in  multiplication,  occur  in  the  work  in  division 
as  "partial  dividends,"  which,  when  combined,  equal  the 
original  dividend.  So  that  the  process  of  division  here 
used  consists  merely  in  breaking  up  the  dividend  into  the 
■component  partial  products,  from  each  one  of  which  ojie 
term  of  the  quotient  is  obtained.  Thus  the  above  work 
is  merely  a  convenient  way  of  breaking  jt:^-f-13ji:+36  into 
the  two  expressions,  (  "  partial  dividends,"  ) 

which  when  written   x{^x-[-A)-\-^{x-[-A) 

•readily  gives  jt:+9  as  the  quotient  by  the  method  given 

in  the  last  exercise. 

60.  We  give  a  few  more  examples  where  multiplication 
and  division  are  exhibited  together,  so  that  the  student 
may  more  clearly  understand  this  method  of  undoing 
multiplication. 

(2)  a  +  1  .  • 

a  +12 
^2+   la 

12^+84 
a+7  )  «2  +  i9^_j:'84  (  a-\-12 

a"-\-  la 

12^  +  84 
12^  +  84 


ARRANGEMENT   OF    WORK.  83 


(3)  2a   -11 

3^    +    4 


6^2— 33« 

8^-44 

2^-11  )  6^2_25a-44  (  3«  +  4 
6^2-33^ 

8^-44 

8«-44 


(4)  a  +9 

a  — 5 


-5^-45 

fl!  +  9  )  ^24.4^  —  45  (dr_5 

a-+9g 

-5^-45 


Some  prefer  to  write  both  divisor  and  quotient  to  the  right  of  the 
dividend.     Thus : 
(5) 


Dividend, 

15x*-44.r+32 

-24X+32 
-24a;-h32 

3.X"— 4     Divisor, 

5x-8     Quotient. 

This  arrangement  saves  space,  and  the  divisor  is  where  it  is  readily 
multiplied  by  each  term  of  the  quotient. 

61.  In  examples  (4)  and  (5)  above  the  student  will 
observ^e  when  the  first  term  of  the  remainder  is  subtractive 
that  the  corresponding  term  found  in  the  quotient  has  the 
opposite  sign  to  the  first  term  of  the  divisor.  Of  course 
the  reason  for  this  is,  that  the  divisor  must  be  multiplied 
by  the  term  of  the  quotient  to  give  the  partial  dividend, 


84  DIVISION. 

and  it  requires  U7ilike  signs  to  give  minus  in  this  multipli- 
cation. If  due  attention  be  given  to  this  fact  no  difficulty 
will  be  found  in  obtaining  the  correct  sign  for  each  term 
of  the  quotient. 

In  this  statement  it  will  be  noticed  that  we  have  spoken  of  a  term 
having  no  sign  at  all  as  if  it  had  the  sign  -|-.     See  Art.  17- 

62.  It  is  very  important  in  the  division  of  a  polynomial 
by  a  polyyioniial  that  both  dividend  and  divisor  be  arranged 
according  to  the  powers  of  a  common  letter.  It  makes  no 
difference  whether  the  arrangement  be  according  to  the 
descending  or  the  ascending  powers  of  a  common  letter, 
but  both  dividend  and  divisor  should  be  arranged  in  the 
sa77ie  order.  Any  letter  may  be  selected  for  this  purpose, 
but  the  letter  which  occurs  the  greatest  number  of  times 
in  the  given  dividend  and  divisor,  is  naturally  preferred. 
If  some  powers  of  the  selected  letter  do  not  occur  in 
the  dividend,  then  it  is  well  to  leave  a  blank  space  in  the 
work  for  every  such  term.  Thus: 
(1)  Divide  a'^—b-  by  a  +  b. 

a-^b)a''-     '     -b'^ia-b 
a'^-\-ab 
-ab-b'^ 
-ab-b"- 


(2)  Also,  divide  a^  —  b^  by  a—b. 

a-b')  a^  —b^  (^a'^+ab+b'^ 

a^-aH 

a'^b 

oH-ab^ 

ab'^-b^ 
ab^^-b^ 


EXAMPLES.  85 


EXERCISE  43. 

Examples. 


Divide 

Divide 

I. 

x-'  +  3jf— 40by  ;r— 5. 

8.  ^2 +4^—45  by  ^—5. 

2. 

X'-\-5x—(^  by  x—1. 

9.  a^—4a^S2  by  a— 8. 

3. 

,r2+7;r— 30by  ;f— 3. 

10.  «2_}_7^_78  by  «— 6. 

4. 

;r2+4;tr-5by;r-l. 

II.  «2_i2i  by  «  +  ll. 

5. 

x''-2x-eShyx-9. 

12.  ;i:2— ^jf— 6^2  by  ;i-— 3«, 

6. 

x^-^7x—Ai  by  :r— 4. 

13.  ;i-2— 9^2  by  A-— 3^. 

7. 

;^2_4by^_2. 

14.  .^-2-49J^'2  by  jr+7)'. 

15.   a^+(yad+dd- 

by  a +  3^. 

16.  a^-—17ad  +  7'2d-hy  a—9d. 

17.  2jr2-9A-+10  1 

by  2a--5. 

18.  3.r2  4-2jt;-l  by  3;*:— 1. 

19.  6j»;2  4-5;»:+21  by  3;t:+7. 

20.  9a'2-64  by  3;f+8.     ' 

21.  9;t:2-3;trj-2j2  by  3^-2^. 

22.  4x^+4xf—3o_y'^  by  2jf+7>'. 

23.  2x'^-^6ax—2Da'^  by  x-^5a. 

24.  4.r-+4«A-f «-  by  2;i-+«. 

25.  oa6  +  15«5+5^  +  15  by  «  +  3. 

5^7+15 

5^-i-ii> 

26.  42«*4-41a^-9«2_9^_l  by  7«2_^8a  +  l. 

7a»-|-8«+l  )  42^^44-4^3—  9rt*-9a— 1  (  6a«-a  — 1 
42(z4+48^34-  6«* 

—  7^?3  — 15rt*  — 9a 

—  1a^—   8«*—   « 


—   7a«-8rt-l 


86  DIVISION. 

27.   Divide  21a^-^U^  by  Za  +  2b. 

27^34-]  8^  V^ 

—  18^2^ 


12.//^  2 +8^  3 

28.  TiWidiQ  a^  +  b^  +  c^  — 2>abc  hy  a  +  b-^c. 
Arrange  according  to  the  descending  powers  of  one  of  the  letters, 
say  a.     It  is  important  to  keep  this  arrangement  throughout  the  work, 
a-\-b-\-c  ]  a^  —  ^abc-\-b^-\-c^  (  a^—ab—ac-\-b^—bc-\-c^ 
a^-\-a^b-\-a^c 

-a^b-a^c  —Zabc 

—a^b  —ab^—  abc 

~         —a^c-\-ab^  —  tabc 

—  a^c  —  abc—ac^ 


ab^—  abc-{-ac^-\-b» 
ab^  _|-^3_j_32^ 

_  abc-\-(ic^         —b^c 

*         —  abr  — b^c—bc^ 

ac^ -^bc^--yc^ 

Divide 

29.  '2a^-^a'^-\-Za'^—Za^-\hya'^  —  Za-^\. 

30.  l7n^—^m^-\-Zm^  —  Zm-\-\hYm'^—^7n^-\. 

31.  G«5^-17a-jr2  +  14«jt3-3jt4  by  2^-3.r. 

32.  4>/i_18>/3+22)/2-7j/+5  by  2y-5. 

33.  4^-^+4a2-29a  +  21  by  2^-3. 

34.  45ji:'*4- 18-^3 +35j»;2+4jtr-4  by  ^x^-^'lx-'l, 

35.  iV>-*«'^+if^'^'+i^^'  by  l-^  +  i^. 

37.x^-\-y^-{-Sxj—lhyx+j/—l. 

38.  «2_2^^  +  ^2_^2^2^^-^-^  by  ^-^+^-^. 


EXAMPLES.  Zj 


Divide 

39.  a^  +  b^-c^-2a''b'' hy  a'^-y^-c'^, 

40.  l+x^+x"^  hy  x'^  +  l—x, 

41.  «-5  — 243  by  «— 3. 

42.  l—Qx'^+bx^hyl  —  2x-\-x'i, 

43.  j»;6-2^3j»;3+a«  byjt:2-2a^4-a2. 

44.  Zx^-Zhy^x-'+^x+i. 


CHAPTER  VII. 
NEGATIVE  QUANTITIES. 
EXERCISE  44. 

Number  and  Quantity. 

63.  Anything  which  can  be  measured  by  a  unit  oi 
the  same  kind  is  called  a  Quantity.  Thus,  10  bushels 
is  a  quantit}^,  the  unit  being  a  bushel,  and  this  unit 
taken  10  times  gives  the  quantity^  10  bushels.  Also  10 
cords  is  a  quantity,  the  unit  in  this  case  being  one  cord, 
and  this  unit  taken  10  times  gives  the  quantity,  10  cords. 

Also  the  abstract  number  10  is  a  quantity,  the  unit  in 
this  case  being  the  abstract  number  1,  and  this  unit  taken 
10  times  gives  the  quantity  10.  Of  course,  the  unit 
itself  is  a  quantity. 

The  word  quantity  as  above  defined,  plainly  includes 
number,  but  while  a  number  is  a  quantity,  a  quantity  is 
not  always  a  number. 

Five  miles  would  be  called  a  quantity  and  never  be 
called  a  number,  but  the  number  5  may  be  called  either  a 
number  or  a  quantity  indifferently. 

The  word  quantity  is  usually  used  as  here  explained, 
but  some  writers  on  Algebra  never  use  the  word  quantity 
to  include  number. 

64.  The  answer  to  a  problem  in  Algebra  is  often 
something  like  5  miles  or  4  tons  or  3  dollars,  or  some 
other  concrete  quantity,  but  the  reasoning  is  always  con- 
ducted by  numbers,  and  so  the  letters  used  in  Algebra! 
always  represent  fiumbers,  and  the  result  reached  is  the 


OPPOSITE    DIRECTIONS.  89 

number  of  miles  or  tons  or  dollars,  or  whatever  it  may 
be,  and  then  the  name  of  the  thing  we  are  considering 
may  be  added  at  the  end  to  the  number  we  have  obtained 
by  working  the  problem. 

EXERCISE  45. 

Opposite  Directions. 

1.  If  a  man  travel  east  30  miles  and  then  west  20 
miles,  how  far  will  he  be  from  the  starting  point  ? 

2.  If  he  travel  east  30  miles  and  then  west  40  miles, 
how  far  will  he  be  from  the  starting  point  ? 

3.  If  the  temperature  is  ?ero,  and  it  rises  10  degrees 
and  then  falls  6  degrees,  how  far  will  it  then  be  from  zero  ? 

4.  If  it  rises  10  degrees  and  then  falls  14  degrees,  how 
far  will  it  then  be  from  zero  ? 

5.  If  a  man  receive  50  dollars  and  spend  35  dollars, 
the  amount  of  money  he  has,  differs  from  what  he  had 
before  by  how  much  ? 

6.  If  he  receive  50  dollars  and  spend  65  dollars,  the 
amount  of  money  he  has,  differs  from  what  he  had  before 
by  how  much  ? 

7.  If  a  man  travel  east  30  miles  and  then  west  20 
miles,  is  he  east  or  west  of  the  starting  point,  and  how 
far? 

8.  If  he  travel  east  30  miles  and  then  west  40  miles,  is 
he  east  or  west  of  the  starting  point,  and  how  far  ? 

9.  If  the  temperature  is  zero  and  it  rises  10  degrees 
and  then  falls  6  degrees,  will  it  then  be  above  or  below 
zero,  and  how  far  ? 

10.  If  the  temperature  is  zero  and  it  rises  10  degrees 
and  then  falls  14  degrees,  will  it  be  above  or  below  zero, 
and  how  far  ? 


90  NEGATIVE    QUANTITIES. 

11.  If  a  man  receive  50  dollars  and  spend  35  dollars, 
has  he  more  or  less  than  he  had  before,  and  how  much  ? 

12.  If  he  receive  50  dollars  and  spend  65  dollars,  has 
he  more  or  less  than  he  had  before,  and  how  much? 

66.  It  is  plain  that  the  last  six  of  these  examples  are 
very  much  like  the  first  six,  and  yet  these  two  sets  differ 
in  one  important  respect.  In  the  first  six  we  are  con- 
cerned only  with  amount,  in  the  last  six  something  be- 
sides amount  is  required,  and  this  something  we  may. 
for  want  of  a  better  word,  call  Direction. 

Illustrations  might  have  been  given  involving  other 
kinds  of  quantity;  as  for  example,  degrees  north  or  so2tth 
of  the  Equator,  longitude  east  or  ivcst,  time  before  or  after 
a  given  event,  gain  or  loss,  etc. 

Plainly  in  those  cases  where  we  consider  direction, 
there  are  two  directions,  either  one  of  which  is  just  the 
reverse,  or  opposite,  of  the  other  one;  but  where  we  are 
concerned  only  with  amount,  there  is  no  such  thing  as 
reverse. 

Time  cannot  be  reversed.  But  time  befoi^e  can  be  re- 
versed, and  the  reverse  of  it  is  time  after. 

Distance  cannot  be  reversed.  But  distance  east  can  be 
reversed,  and  the  reverse  of  it  is  distance  west,  etc. 

EXERCISE  46. 

How  Directions  are  Distinguished. 

66.  Sometimes  in  Algebra  w^e  are  not  called  upon  to 
consider  anything  but  amount,  or  magnitude,  and  some- 
times we  are  obliged  to  consider,  both  magnitude  and 
direction,  and  so  we  must  in  some  way  distinguish  be- 
tween the  two  opposite  directions.  To  find  out  how  to 
distinguish  between  these  opposite  directions,  let  us  con- 
sider the  following  examples. 


HOW    DIRECTIONS   ARE    DISTINGUISHED.  9 1 

The  distinction  between  positive  and  negative  is  made 
by  means  of  the  signs  -f  and  — .  To  explain  this  let  us 
consider  tv/o  or  three  questions  : 

1.  If  a  man  has  $1000,  and  he  gains  $500,  and  then 
looses  1700,  how  much  will  he  then  have  ?  How  much 
less  is  IOOO  +  0OO-7OO  than  1000  ^ 

2.  A  man  had  a  dollars  and  gained  500  dollars,  and 
then  lost  700  dollars,  how  much  did  he  then  have  ?  How 
much  less  is  «4-o00— 700  than  «?  Write  an  expression 
of  two  terms  which  shall  be  equal  to  ^  +  oOO— 700. 

The  two  terms  +500—700  can  be  replaced  by  what 
single  term  ? 

3.  A  man,  100  miles  east  of  St.  I,ouis,  travels  east  50 
miles  and  then  west  75  miles,  how  far  east  ot  St.  Louis 
is  he  then  ?  A  man,  a  miles  east  ot  St.  Louis,  travels  50 
miles  east  and  75  miles  west,  how  far  is  he  then  east  of 
St.  Louis  ?  Is  his  distance  east  expressed  by  a -1-50—75  ? 
Is  it  also  expressed  by  a— 25  ? 

67.  Notice  in  these  examples  that  two  terms,  one 
additive  and  one  subtractive,  may  be  replaced  by  a  single 
term  which  in  each  example  given  has  been  a  subtractive 
term.  Notice  also,  that  in  the  expressions  with  three 
terms,  the  additive  and  subtractive  terms  express  oppo- 
site directions,  as  in  the  last  example  the  +  sign  was 
used  in  connection  with  the  distance  traveled  east,  and 
the  —  sign  used  in  connection  with  the  distance  traveled 
west. 

We  are  naturally  led  to  associate  the  +  sign  %vith  one 
direction  and  the  —  sign  with  the  other  direction. 

In  example  8,  we  could  replace  +50—75  b}^  —25. 
Now  we  know  that  a  journey  of  50  miles  east,  followed 
by  one  of  75  miles  west,  is  equivalent  to  a  single  journey 


92  NEGATIVE    QUANTITIES. 

of  25  miles  west,  so  the  term  —25  which  could  be  substi- 
tuted for  +50—75  comes  to  have  a  meaning,  namely  25 
miles  west  in  this  example. 

In  example  2,  we  could  replace  the  two  terms  -^-500 
—700  by  the  single  term  —200,  and  here,  also,  a  gain  of 
$500  followed  by  a  loss  oi  $700,  is  the  same  as  a  single 
loss  of  $200,  so  that  the  term  —200,  which  could  be  sub- 
stituted for  +  500—700,  comes  to  have  a  meaning,  namely, 
$200  loss  in  this  example. 

Many  other  illustrations  might  be  given,  but  these  are 
enough  to  show  that  we  distmgidsh  between  the  two  oppo- 
site directions  by  means  of  the  signs  +  arid  — . 

We  further  .see  that  when  the  distinction  is  so  made, 
each  term  of  the  expressio7i  co7nes  to  have  a  meariing^  and 
so  it  does  not  ?natter  which  term  is  written  first. 

68.  We  see  from  this  that  the  Signs  Plus  and 
Minus  have  a  Double  Use  in  Algebra;  first,  to  indicate 
the  operations  oi  addition  and  subtraction  respectively; 
-second,  to  distinguish  between  opposite  directions  as  just 
described.  There  is  no  danger  of  confusion  arising  from 
this  double  use;  the  context  will  always  make  it  clear  in 
which  sense  the  signs  are  used. 

EXERCISE  47. 

Positive  and  Negative  Numbers. 

69.  In  Arithmetic  we  are  concerned  only  with  the 
numbers  0,     1,     2,     3,     4,     etc., 

and  intermediate  numbers  ;  but  in  Algebra  we  consider 
besides  these  the  numbers 

0,   -1,   -2,   -3,   -4,    etc., 
and  intermediate  numbers. 


POSITIVE   AND    NEGATIVE    NUMBERS.  93 

We  may  represent  the  two  classes  of  numbers  con- 
sidered in  Algebra  on  the  following  scale  : 
.  .  -5,  -4,  -3,  -2,  -1,  0,  1,  2,  3,  4,  5,  .  . 
which  extends  indefinitely  in  both  directions  from  zero. 
The  sign  -f-  perhaps  ought  to  precede  each  of  the  num- 
bers at  the  right  of  zero  on  this  scale,  but  we  will  agree 
that  when  no  sign  is  written  before  a  number  the  sign  -}- 
is  understood. 

70.  Numbers  to  the  right  of  zero  in  the  above  scale 
are  called  Positive,  and  those  to  the  left  of  zero  are  called 
Negative,  or,  we  might  say,*  numbers  represented  by 
figures  preceded  by  a  +  sign  or  no  sign  at  all  are  positive 
and  numbers  represented  by  figures  preceded  by  a  — 
sign  are  negative. 

71.  In  Algebra  numbers  are  often  represented  by 
letters,  and  we  have  already  seen  that  a  letter  may  stand 
for  a  whole  number  or  a  fraction.  We  will  now  further 
extend  the  signification  of  a  letter  by  allowing  that  it 
may  stand  for  one  of  the  numbers  to  the  left  of  zero  in 
the  above  scale  as  well  as  one  to  the  right  of  zero  ;  so 
that  while  in  the  case  of  a  number  represented  by  figures 
we  can  tell  whether  the  number  is  positive  or  negative  by 
the  sign  that  precedes  it,  yet  in  the  case  of  a  number 
represented  by  a  letter,  we  cannot  tell  by  the  sign  before 
it  whether  it  is  positive  or  negative. 

A  minus  sign  before  a  number  always  represeyits  a  7121771- 
ber  of  the  opposite  kind  from  that  7'epresented  by  the  sa7ne 
number  ivith  a  plus  sig7i  or  710  sign  at  all  before  it. 

We  know  that  the  number  5  is  positive,  but  we  do  not 
know  whether  a  is  positive  or  negative  until  we  know 
its  value. 


94  NEGATIVE    QUANTITIES. 

We  know  that  —  5  is  negative,  but  we  do  not  know  whether 

— «  is  positive  or  negative  until  we  know  the  value  of  a. 

If  «=3,  then  — «=— 3,  and  a  is  positive  and  —a  is 

negative;  but  if  «=— 3,  then  — «=3  (because  3  is  the 

opposite  of  —3),  and  a  is  negative  and  — «  is  positive. 

72.  \^ie  have  seen  that  we  are  required  to  distinguish 
between  quantities  opposite  to  each  other  and  that  this 
distinction  is  made  by  means  of  the  signs  plus  and  minus; 
for  example,  if  +10  degrees  means  a  temperature  of  10 
degrees  above  zero,  then  —10  degreed  would  mean  a 
temperature  of  10  degree  below  zero,  and  if  +10  miles 
means  10  miles  north  of  the  equator,  then  —10  miles 
would  mean  10  miles  south  of  the  equator,  and  if  +10 
rods  means  10  rods  east  of  a  given  point,  then  —10  rods 
would  mean  10  rods  west  of  the  same  given  point,  and 
if  +10  be  10  units  of  miy  kind  in  any  sense,  then  —10 
would  be  10  units  of  the  same  kind  in  just  the  opposite  s&nso:. 

In  each  case  one  of  these  quantities  is  positive  and  the 
opposite  one  is  negative.  Either  direction  may  be  selected 
as  positive,  and  then,  of  course,  the  opposite  direction 
will  be  negative.  But  it  is  almost  alwa3'S  easiest  and 
best  to  select  the  quantity  about  which  we  are  inquiring 
in  any  given  problem  as  positive. 

Anything  that  increases  the  quantity  we  have  chosen 
as  positive  will  also  be  positive,  and  anything  that  de- 
creases our  selected  positive  quantity  will  be  negative. 

EXERCISE  48. 

Illustrative  Examples. 

I.  One  day  a  man  travels  east  100  miles,  the  next  day 
he  travels  west  250  miles,  and  the  third  day  he  travels 
east  175  miles;  how  far  is  he  then  east  of  his  starting 
point  ? 


ILLUSTRATIVE    EXAMPLES.  95 

SOLUTION. 

because  the  problem  asks  how  far  east  the  man  is  of  his  starting 
point,  we  take  distance  east  to  be  positive,  and  therefore,  distance 
west  to  be  negative.  We  associate  distance  east  with  the  -)-  sign,  and 
distance  west  with  the  —  sign.  The  distance  the  man  is  east  of  his 
starting  point  will,  therefore,  be  represented  by 

100-250+175 
which  is  equal  to  25.     Hence,  the  man  is  25  miles  east  of  his  starting 
point. 

2.  One  day  a  man  travels  east  100  miles,  next  day  he 
travels  west  250  miles,  and  the  third  day  he  travels  east 
50  miles;  how  far  east  is  he  then  from  his  starting  point? 

SOLUTION. 

Because  the  problem  asks  how  far  east  the  man  is  of  his  starting 
point,  we  take  as  before  distance  east  to  be  positive,  and  of  course, 
distance  west  to  be  negative,  and  asssociate  distance  east  with  the  -)- 
sign  and  distance  west  with  the  —  sign. 

The  distance  the  man  is  east  of  his  starting  point  will  then  be  rep- 
resented by  100  -  250-f-50. 

Now,  these  three  terms  may  be'  replaced  by  —100,  hence  we  say 
100-250+50=  —  100.  Therefore,  the  distance  from  the  starting  point 
is  represented  by  —100,  which  by  our  interpretation  of  negative  quan- 
tities means  100  miles  7vest  of  the  starting  point. 

SECOND    SOLUTION. 

Let  us  now  take  distance  ivest  to  be  positive,  and  therefore  distance 
east  negative,  and  everywhere  associate  distance  west  with  the  + 
sign,  and  distance  east  with  the  —  sign.  Upon  this  assumption  the 
distance  the  man  is  west  of  his  starting  point  will  be  represented  by 

—  100+250-50 
which  is  equal  to  100.     Hence,  as  before,  the  man  is  100  miles  west  of 
his  starting  point. 

3.  A  boy  rows  his  boat  up  stream  3  miles  in  an  hour, 
and  then  rests  an  hour,  when  he  floats  down  stream  5 
miles,  he  then  rows  up  stream  again  3  miles  in  the  next 
hour;  how  far  is  he  above  the  starting  point  ? 

Between  what  two  directions  are  we  required  to  distinguish  in  this 
problem  ?  Which  is  it  most  natural  to  take  for  the  positive  direction, 
up  stream  or  down  stream  ? 


g6  NEGATIVE    QUANTITIES. 

4.  A  boy  rows  his  boat  up  stream  3  miles  in  an  hour 
and  then  rests  an  hour,  when  he  floats  down  stream  6 
miles,  he  then  rows  up  stream  again  2  miles  in  the  next 
hour;  how  far  is  he  above  his  starting  point? 

5.  A  boy  rows  his  boat  up  stream  3  miles  in  an  hour 
and  then  re.ots  an  hour,  when  he  floats  down  stream  6 
miles,  he  then  rows  up  stream  again  2  miles  in  the  next 
hour;  how  far  is  he  then  below  his  starting  point  ? 

6.  A  merchant  was  in  business  3  years;  the  first  year 
he  lost  $2000,  the  second  year  he  gained  $500,  and  the 
third  year  he  gained  $2600.  What  was  his  profit  in 
business  ? 

7.  A  merchant  was  in  business  3  years;  the  first  year 
he  lost  $3000,  the  second  year  he  gained  $500,  and  the 
third  year  he  gained  $2000.  What  was  his  profit  in 
business  ? 

8.  A  merchant  was  in  business  3  years;  the  first  year 
he  lost  $3000,  the  second  year  he  lost  $500,  and  the  third 
year  he  gained  $2500.     What  was  his  profit  in  business  ? 

9.  A  merchant  was  in  business  3  years;  the  first  year 
he  lost  $3000,  the  second  year  he  lost  $500,  and  the  third 
year  he  gained  $2500.     What  was  his  loss  in  business  ? 

10.  A  merchant  was  in  business  4  years;  the  first 
year  he  lost  $2550,  the  second  year  he  gained  $1025,  the 
third  year  he  gained  $575,  and  the  fourth  year  he  gained 
$900.     Did  he  gain  or  lose,  and  how  much  ? 

11.  On  a  cold  morning  the  temperature  was  10  degrees 
below  zero,  and  it  rose  12  degrees  during  the  day;  what 
was  the  temperature  at  evening  ? 

12.  On  a  cold  morning  the  temperature  was  10  degrees 
below  zero,  and  it  rose  12  degrees  during  the  day,  and 
fell  6  degrees  during  the  night;  what  was  the  temperature 
the  next  morning  ? 


negative  quantities  in  addition.         97 

73.  Fundamental  Operations  with  Negative 
Numbers.  Until  the  beginning  of  the  present  chapter  the 
letters  used  always  stood  for  positive  numbers,  but  now 
a  letter  may  stand  for  either  a  positive  or  a  negative 
number.  Moreover,  a  letter  standing  alone  with  a  minus 
sign  before  it  has  a  meaning  now,  but  to  know  the  mean- 
ing of  a  letter  with  a  minus  sign  before  it,  as,  for  instance, 
—a,  we  must  first  know  what  a  means  when  preceded  by 
a  -f  sign  or  by  no  sign  at  all,  and  then  give  to  —a  the 
opposite  interpretation.  As  letters  now  have  a  much 
larger  significance  than  before,  we  must  briefly  re-examine 
the  four  fundamental  operations  of  Algebra,  viz.:  Ad- 
dition, Subtraction,  Multiplication,  and  Division,  to  see 
if  the  results  previously  obtained,  on  the  supposition  that 
the  letters  stood  for  positive  numbers,  hold  when  the 
letters  stand  for  negative  as  well  as  for  positive  numbers. 

EXERCISE  49. 

Addition. 

74.  Addition  is  the  process  of  finding  the  result  of 
two  or  more  numbers  taken  together.  The  result  found 
is  the  sum. 

This  definition  agrees  with  all  before  found,  when  all 
the  numbers  used  are  positive,  and,  as  we  shall  see,  is  suf- 
ficiently broad  to  include  the  case  of  negative  numbers. 

If  we  wish  to  add  -f  25  and  —15  together,  we  get  for 
our  result  -f  25-15  or +10. 

This  is  called  adding  —15  to  -|-25,  and  is  easily  seen 
to  be  the  same  as  subtracting  -fl5  from  +25. 

With  this  definition  of  addition,  it  is  easy  to  see  that 
to  find  the  sum  of  two  or  more  numbers,  we  supply  + 
signs  to  terms  having  no  signs,  and  then,  write  them 
down  with  signs  unchanged,  and  combine  terms  if  pos- 
sible, the  same  as  in  Chapter  III. 
7 


98  NEGATIVE    QUANTITIES. 

To  add  7,  5,  and  —8  we  supply  +  signs  before  the 
terms  7  and  5,  and  write 

+  7  +  5-8. 

When  the  sign  of  the  first  term  is  +  we  may  omit  the 
sign  if  we  like,  but  we  must  never  omit  the  —  sign,  and 
never  the  +  sign  in  any  term  but  the  first. 

Addition  is  nicely  illustrated  by  the  scale  of  algebraic 
numbers.  For  example,  if  we  wish  to  add  6  to  —4  we 
begin  at  —4  on  the  scale  and  count  forward  6  spaces, 
arriving  thereby  at  2.  Again,  to  add  —6  to  —4,  we 
begin  at  —4  on  the  scale  and  count  backward  6  spaces, 
arriving  thereby  at  —10. 

Now,  if  we  use  letters,  we  can  easily  find  the  sum  by 
the  method  here  given. 

Suppose  we  wish  to  add  ^,  — ^,and  —c.  Supplying  + 
sign  before  a,  and  writing  down  with  signs  unchanged 
we  get,  -\-a—b—c. 

Suppose  next  we  wish  to  add  a-\-b,  b—c,  and^— /. 

Supplying  +  signs  before  terms  <2,  b^  and  ^,  and  writing 
down  with  signs  unchanged  we  get, 

-\-a-\-b-\-b—c-\-e—f 
or  a-\-2b—c-\-e—f. 

Exactly  the  result  obtained  by  the  method  of  Chapter  III. 

Any  result  reached  in  Chapter  III  could  easily  be  tested 
by  the  notion  of  addition  here  presented,  and  found  cor- 
rect. Therefore,  the  methods  and  results  of  Chapter  III 
hold,  whether  the  numbers  used  are  positive  or  negative. 

One  important  difference  may  now  be  noticed  between 
addition  in  Arithmetic  and  Algebra.  In  Arithmetic  ad- 
dition implies  augmentation,  but  in  Algebra  this  is  not 
necessarily  the  case. 


NEGATIVE    QUANTITIES    IN    SUBTRACTION.         99 

Examples. 

1.  Add  a-\-d-\-c,  —Za—b-\-c,  and  —a—b—4iC. 

2.  Add  —r—2s—Zt,  —r-^t,  and  — 2r— 2^. 

3.  Add  a—b-\-c,  —a-\-b-\-c,  and  a-\-b—c. 

4.  Add«4-^,  — 1^,  — 1^,  and  4^. 

5.  If  x=a-\-b—'2c,  y=a—2b+c,  and  2'=— 2a  +  ^+^ 
show  that  x+y-\-2=0. 

6.  If  x=a  +  b—c,  jy=za—b-\-Cy  and  ^=--a4-^+^  show 
that  x-}-jy-i-2=a-\-b-\-c. 

EXERCISE  50. 

Subtraction. 

75.  Subtraction  is  the  reverse  of  addition  /.  e.j  it  is 
the  process  of  finding  a  number  called  the  remainder, 
which  added  to  the  subtrahend  will  give  the  minuend. 
For  example,  12  is  a  number,  which  added  to  —5  will 
give  7;  therefore,  —5  subtracted  from  7  gives  12. 

It  is  easy  to  see  from  this  meaning  of  subtraction  that 
to  subtract  one  number  from  another,  we  chayige  the  sign 
of  the  subiraheiid  and  then  unite  it  to  the  7ninuend. 

This  is  exactly  the  method  of  Chapter  IV  ;  therefore, 
the  methods  and  results  of  Chapter  IV  hold,  when  the 
letters  used  stand  for  negative  as  well  as  positive  numbers. 

Examples. 

1.  From  jr2+jj/2  take  x'^—y'^. 

2.  From  x-\-a—W^  take  —x—Sa  +  b^, 

3.  From  «--^4-<^— ^take  ^z+^— ^4-^. 

4.  From  2?i^+Sa^  —  r^—s^  take  n^—a^-\-r^—2s^. 

5.  From  ^24. 2^3+^2  take  a^-2ab-hb\ 

6.  'Brova.  uvw'^^2tiv^w-\-S2i^vw  take   Suvw'^-{-2uv'^w 


lOO  NEGATIVE   QUANTITIES. 

7.  From  the  sum  of  a^-{-d^  and  —2ad  subtract  the  sum 
ofa2— ^2and3^3 

8.  From  x^ -^-ax^ -^a'^x-}-a^  subtract  2ax'^—a^x,  and 
from  this  difference  subtract  2ax'^—a'^x. 

9.  What  must  be  added  to  r^-\-s^  +  t^  to  produce  3  ? 

10.  What  must  be  subtracted  from  adc^  to  produce  m-\-r? 

11.  What  must  9ad  be  subtracted  from  to  produce  —ad? 

EXERCISE  51. 

Multiplication. 

76.  The  definition  of  multiplication  given  in  Chapter 
V  is  sufficient  to  include  the  case  of  negative  numbers. 

What  is  the  product  of  6  multiplied  by  —  3  ?  To  pro- 
duce —3  from  unity  we  take  unity  three  times  and  reverse 
the  result.  Hence  to  multiply  6  by  —3  we  must  take  6 
three  times  and  reverse  the  result,  giving  —18. 

What  is  the  product  of  —6  multiplied  by  3  ?  To  pro- 
duce 3  from  unity  we  take  unity  3  times  ;  hence  to  mul- 
tiply —6  by  3  we  take  —6  three  times,  giving  —18. 

What  is  the  product  of  —6  multiplied  by  —  3  ?  To  pro- 
duce —3  from  unity  we  take  unity  3  times  and  then  reverse 
the  result ;  hence  to  multiply  —6  by  —3  we  take  —6  three 
times,  giving  —18,  and  reverse  the  result,  giving  -f  18. 

From  these  illustrations  it  is  evident  that  numbers  can 
be  multiplied  as  in  Chapter  V  whether  the  letters  used 
stand  for  positive  or  negative  numbers.  It  is  also  evident 
that  we  have  here  another  and  very  nice  demonstration 
of  the  ''Law  0/  Signs''  in  multiplication. 

Examples. 
Find  the  product  of 

1.  10  multiplied  by  —4.     3.  —| multiplied  by  — y\. 

2.  —-25  multiplied  by  4.     4.  adc multipYied  by —a^d'h. 


NEGATIVE    QUANTITIES    IN    DIVISION.  lOI 

5.  —fms^  multiplied  by  ^m^s. 

6.  a^-\-d^  multiplied  by  —a^  +  d^. 

7.  —habc—Zrs  multiplied  by  f)abc—Zrs. 

8.  —«—^—r  multiplied  by —a4-33— 9^. 

9.  5^3  +  6^3  multiplied  by  — 7r+25— /. 

10.   —x^y^—xz^-\-yz^  multiplied  by  ^xy—\y^, 

EXERCISE  52. 

Division. 

77.     Division  is  the  reverse  of  multiplication,  that  is, 
it  is  the  process  of  finding  a  number  called  the  quotient, 
which  multiplied  by  the  divisor,   equals  the  dividend. 
We  may  express  this  in  the  form  of  an  equation  thus, 
Divisor  X  Quotieiit  =  Divideiid. 

As  here  arranged  this  is  a  case  of  multiplication  where 
the  divisor  is  the  multiplicand,  the  quotient  is  the  multi- 
plier, and  the  dividend  is  the  product.  As  we  know  in 
multiplication,  that  when  the  multiplier  has  the  sign  + 
the  signs  of  the  multiplicand  and  product  are  alike,  it 
follows  here  that  when  the  quotient  has  the  sign  -f  the 
dividend  and  divisor  must  have  like  signs,  or  stated  in 
the  reverse  order,  when  the  signs  of  the  dividend  ayid 
divisor  are  alike  the  sign  of  the  quotient  is  + . 

It  is  also  easy  to  see  that  when  the  quotient  has  the 
sign  —  the  signs  of  the  dividend  and  divisor  are  unlike, 
or  stated  in  reverse  order.  Whe^i  the  signs  of  the  dividend 
and  divisor  are  imlike^  the  sign  of  the  qnotieyit  is  — . 

These  two  statements  put  together  give  the  Law  of 
Signs  in  Division,  viz.:  In  division  like  signs  give  +  and 
unlike  sig7is  give  — . 


102  NEGATIVE   QUANTITIES. 

Examples, 

1.  Divide  10  by  —5.  5.  Divide  —63  by  —7. 

2.  Divide  5  by  —10.  6.  Divide  a}  by  —a. 

3.  Divide  27  by  —3.  7.  Divide  aH^  by  — ^*. 

4.  Divide  —63  by  7.  8.  Divide  —aH^c^  by  a^b'^c 

9.  Divide  m^r^s'^  by  ^ms^. 
10.  Divide  6^3__^2_i4^^3  ^^^  3a2-f-4«— 1. 


CHAPTER  VIII. 
PARENTHESES. 

EXERCISE  53. 

Removal  of  Parentheses. 

78.  The  subject  of  parentheses  has  already  been  con- 
sidered to  some  extent,  and  we  have  already  learned  that 
an  expression  within  a  parenthesis  is  to  be  looked  upon 
as  a  single  number  just  as  though  it  were  represented  by 
a  single  symbol. 

Now,  it  may  happen  that  an  expression  within  a  paren- 
thesis is  itself  an  expression  which  contain^  a  parenthesis, 
so  we  would  have  a  parenthesis  within  a  parenthesis.  In- 
deed, we  may  have  several  parentheses  one  within  another. 

These  complicated  expressions  present  no  difficulty,  for 
we  can  take  the  parentheses  one  at  a  time,  and  if  w^e  know 
how  to  remove  one,  we  may  do  this  and  then  remove  an- 
other, and  so  on  until  all  are  removed.  For  example,  if 
we  wish  to  remove  the  parentheses  from 

we  begin  by  removing  the  inner  parenthesis  first  and 
write  the  expression  in  the  form 

and  now  by  removing  the  remaining  parenthesis  we  write 
the  result  in  the  final  form 

a-\-b—c-\-d-\-e, 

79.  In  removing  parentheses  it  is  usually  best  to  re- 
move the  innermost  parenthesis  first,  and  then  the  inner^ 
most  parenthesis  of  all  that  remains,  and  so  on  until  all, 
or  as  many  as  may  be  desired,  are  removed. 


I04  PARENTHESES. 

80.  When  several  parentheses  are  used  one  within 
another,  they  are  often  made  of  different  shapes  and 
sometimes  of  different  sizes  to  prevent  confusion.  Some 
of  the  forms  used  are,  ()>{}»[]•  Sometimes  a  hori- 
zontal line,  called  a  Vinculum  or  bar,  is  drawn  above  an 
expression  instead  of  using  a  parenthesis.     Thus, 

means  the  same  as  <2-f(;tr—jj/). 

Remove  the  parentheses  from  the  following  expressions : 

1.  a-\-[d-ic-\-d)']. 

2.  a-^ld—(c-{-d—e)-{-2']. 

3.  Sa-^2d-li5d^-4)-(U-2)l 

4.  w2+«2_(^3_|-^3^5(2:r-fl)]-r). 

5.  4;r-f3>/-5;tr-[2)/-(6;r-6>/)]. 

6.  4a-(Qa-l5a-(ia-2a)'\). 

7.  Sx-(4y-l6x-(Qy-7x)]). 

8.  Sx+i-4:yj-l5x-C6y-7x}]). 

9.  5a^-(4d^-[S(_a^  +  d'')-4.(x-2)])-{-2. 

EXERCISE  54. 

Insertion  of  Parentheses. 

1.  If  the  parenthesis  be  removed  from  a  +  {d—c-{-d), 
what  is  the  result  ? 

2.  If  the  parenthesis  be  removed  from  a  —  (^—b-\-c—ci) 
what  is  the  result  ? 

Notice  how  the  expressions  within  the  parentheses  in  these  two 
examples  are  related,  and  notice,  also,  how  the  results  are  related. 

3.  If  from  any  expression  a  parenthesis  preceded  by  a 
-f  sign  be  removed,  what  is  the  effect  on  the  terms 
originally  within  the  parenthesis  ? 


INSERTION    OF    PARENTHESES.  I05 

4.  How,  therefore,  can  any  number  oi  terms  be  brought 
within  a  parenthesis,  preceded  by  a  -f  sign  ? 

5.  If  from  any  expression  a  parenthesis  preceded  by  a 
-—  sign  be  removed,  what  is  the  effect  on  the  terms  orig- 
inally within  the  parenthesis  ? 

6.  How,  therefore,  can  any  number  of  terms  be  brought 
within  a  parenthesis,  preceded  by  a  —  sign  ? 

7.  Enclose  the  last  three  terms  of  a^-\-d^^c'^-\-5  in  a 
parenthesis,  preceded  by  a  +  sign. 

8.  Enclose  the  last  three  terms  of  a^  +  b^  — ^*  +  5  within 
a  parenthesis,  preceded  by  a  —  sign. 

9.  Enclose  the  last  two  terms  of  a-  — /5^— c*+5  within 
a  parenthesis,  preceded  by  a  +  sign. 

10.  Enclose  the  last  two  terms  of  a'^-\-b^—c^-\-b  within 
a  parenthesis,  preceded  by  a  —  sign. 

11.  Enclose  the  last  three  terms  of  a-\-b—Ac—^e'^-\-Qr^ 
— w*  — 16  within  a  parenthesis,  preceded  by  a  +  sign. 

12.  Enclose  all  but  the  first  term  of  a-f  ^— 4<:-f5^^-f  6r^ 
— ;2*  — 16  within  a  parenthesis,  preceded  by  a  —  sign. 

13.  Enclose  the  third  and  fourtli  terms  of  a-\-b—Ac-{-be'^ 
+  6r^— w^  — 16  within  a  parenthesis,  preceded  by  a  — 
sign,  and  the  fifth,  sixth,  and  seventh  terms  of  the  same 
expression  in  another  parenthesis,  preceded  by  a  —  sign. 

14.  Fill  out  the  blank  parenthesis  in  the  equation 
«2  +  (2^_l)=a2-(         ). 

15.  Fill  out  the  blank  parenthesis  in  the  equation 
aH-\bcd-Zx''-Qt^-Si)'\=a''b+(^        )  +  7/*-9. 


CHAPTER  IX.- 

ELEMENTARY  FACTORS,  MULTIPLES, 
AND  FRACTIONS. 

EXERCISE  55. 

Factors. 

81.  A  definition  of  factor  has  been  given,  (see  page 
12),  and  we  have  already  learned  that  an  expression  may 
have  several  factors.  For  example,  the  different  factors 
of  lO;^^  are 

2,  5,  10,  X,  2x,  5x,  10;r,  x\  2x^,  5x^. 
Of  these  factors  2,  5  and  x  may  be  called  Prime,  because 
they  cannot  be  further  factored. 

The  expression  lO;^^  contains  the  prime  factor  x  twice, 
so  all  the  prime  factors  of  IOjt^  are  2,  5,  x,  x;  and  as 
any  expression  equals  the  product  of  all  its  prime  factors, 
we  have 

10ji:2  =  2x5;r;i;. 

When  an  expression  is  written  as  the  product  of  all  its 
prime  factors,  it  is  said  to  be  Resolved  into  its  Prime 
Factors. 

Resolve  the  following  eight  expressions  into  their 
prime  factors : 

1.  SOx^y\     3.  SSad^c^.      5.  ISSr^-s^.     7.  2431uv^w^, 

2.  150^5^^    4.  51m^r\      6.  42dadn.      8.  25dx^j;2*, 
9.  Find  three  of  the  factors  BOx'^y^  which  are  not  prime. 

10.  What  are  the  different  prime  factors  of  15a  ^  ? 

11.  What  are  all  the  prime  factors  of  \ha^  ? 

12.  Resolve  %a^b^c^  into  its  prime  factors. 


HIGHEST    COMMON    FACTOR.  lO/ 

EXERCISE  56. 

Highest  Common  Factor. 

1.  Does  the  factor  a  belong  to  each  of  the  two  expres- 
sions ha'^-x  and  lax"""  ?  Does  the  factor  x  belong  to  each 
of  these  two  expressions  ?  Does  the  factor  ax  belong  to 
each  expression? 

When  the  same  factor  belongs  to  two  or  more  expres- 
sions it  is  called  a  Common  Factor  of  those  expressions. 

When  a  -brime  factor  is  common  to  two  or  more  ex- 
pressions it  is  called  a  Common  Prime  Factor  of  those 
expressions. 

2.  If  2  and  3  are  each  factors  of  an  expression,  is  6  a 
factor  of  that  expression  ? 

3.  If  a  and  b  are  each  factors  of  an  expression,  is  ab  a 
factor  of  that  expression  ? 

'  4.  If  a  and  b  are  each  comnioyi  factors  of  two  or  more 
expressions,  is  ab  a  common  factor  of  those  expressions  ? 
5.  Is  the  product  of  any  number  of  common  factors  of 
two  or  more  expressions,  a  common  factor  of  those  ex- 
pressions ? 

82.  The  product  of  all  the  common  prime  factors  of 
two  or  more  expressions  is  called  the  Highest  Common 
Factor  of  those  expressions.  The  abbreviation  H.  C.  F. 
is  frequently  used  to  stand  for  the  highest  common  factor. 

83.  From  the  definition  of  the  H.  C.  F.  it  follows  at 
once  that  the  way  to  find  H.  C.  F.  of  two  or  more  expres- 
sions, is  to  resolve  each  expressioti  into  its  prime  factors, 
afid  take  the  product  0/  all  those  which  are  coifmioji  to  all  the 
expressions. 


I08         FACTORS,   MULTIPLES,   AND    FRACTIONS. 

Find  the  H.  C.  F.  of  the  following  expressions  ; 

1.  x^jy  and  x^j'^.  5.  7uvw  and  lOv^wx"^. 

2.  ^abc^  and  lla^b.         6.  x^y,     ^xyz  and  lOx'^2-. 

3.  6r^2  and  15r-^/^.        7.  Zx'^yz,    Ibxzw^  and  \2xyw. 

4.  la'^bc^  and  14a^f^.     8.  4mn^,   Smn^'x,  and  12:r?;2;?® 

9.  50^3^3^%    75aHcd,  and  80  a^M. 

10.  5^5*/^    Tr^^jt:*,    9r2;i;3,   and   Urs^x. 

11.  35;»;3jK,    63aJr2^^   and  70abxy2. 

12.  17«5^6rV8,    51;;z3a*/^3^5^S   and    Sira^d^c^dK 

13.  ^2jK^,    AT^/^^,    ;rj'2'2,    and  x'^y'^z'^. 

14.  2rs^uv,    Zr'^stu,    A.rs'^tv,   and  Sr^^y^/ze^^. 

15.  lu^v^w'^ ,     S5m?iu'^v^y     ISQr^u^v^w,    and   77iiv^w. 

16.  21«3^4^6^8,     63^9/^7c6^jj/-3',     84«2^3^52^^^^     and 
49«6^V;r. 

17.  99;t:2,     187^^jK,     2537<^,     143^2;,  and  275;*;*. 

18.  2a^-n^x^,     12bn^y,     lOOc^fz^x,  and   Ae^i^xy^. 

19.  IGa^^jr^,     SOb'^ny^,     ZQabr^x\  and   28^^r^^ 

20.  lZx^y^z\     dOx^yz,     Uxy\     360>/*^,  and   ZQxs. 

EXERCISE  57. 

Lowest  Common  Multiple. 

84.  When  any  number  or  expression  is  multiplied 
by  something,  the  product  is  called  a  Multiple  of  the 
given  number  or  expression.  Thus,  10  is  a  multiple  of  5, 
50  is  another  multiple  of  5,  100  still  another  multiple  of 
5,  etc.;  10  is  also  a  multiple  of  2  as  well  as  of  5,  50  is  a 
multiple  of  25  as  well  as  of  5,  etc. 


LOWEST   COMMON    MULTIPLE.  IO9 

1.  Is  100  a  multiple  of  5  ?  Is  100  a  multiple  of  10? 
Is  100  a  multiple  of  2  ? 

2.  Is  2^2  a  multiple  of  2  ?     Of  «  ?     Oi  a^  ? 

3.  Is  «<^2^-  a  multiple  of  a?  Of  ^?  Of  ;tr  ?  Of  «^  ? 
Of«^?     Of  adx? 

85.  From  these  questions  it  is  evident  that  the  same 
expression  may  be  a  multiple  of  several  different  expres- 
sions, in  which  case  it  is  called  a  Common  Multiple  ot 
those  expressions. 

4.  Is  ad'^x  a  common  multiple  of  ad,  ax  and  abx  ? 

5.  Is  a'^bx  also  a  common  multiple  of  ab,  ax  and  abx"? 
« 

6.  Is  a^b^x'^  Si  common  multiple  of  the  same  three  ex- 
pressions ? 

86.  From  what  is  here  given,  it  is  plain  that  two  or 
more  given  expressions  may  have  more  than  one  com- 
mon multiple.  Indeed,  if  any  common  multiple  of  two 
or  more  given  expressions  be  found,  then  if  this  common 
multiple  be  multiplied  by  any  number  whatever,  the 
result  will  also  be  a  common  multiple  of  the  given 
expression. 

Any  common  multiple  of  two  or  more  expressions 
contains  all  the  prime  factors  of  each  of  the  given  ex- 
pressions. 

That  common  multiple  which  contains  the  leasf  number 
of  prime  factors  is  called  the  Lowest  Common  Multi- 
ple of  the  given  expressions.  The  abbreviation  I^.  C. 
M.  is  frequently  used  to  stand  for  the  lowest  common 
multiple. 


no        FACTORS,   MULTIPLES,   AND    FRACTIONS. 

87.  From  what  we  have  had  it  follows  that  to  find 
the  L.  C.  M.  of  two  or  more  expressions  we  proceed  as 
follows  : 

Resolve  each  expressioji  mto  its  prime  factors  and  form  a 
product  in  which  each  of  these  prime  factors  occurs  as  many 
times  as  it  occurs  in  that  one  of  the  given  expressio7is  in 
which  it  occurs  the  greatest  7iumber  of  times. 

Find  the  L.  C.  M.  oiZa'^x'^y  and  ZOax'^, 

Sa^x^y  —  'daaxxy; 
'60ax^  =  ^X2X5axx. 
The  prime  factors  are  2,  3,  5,  a,  x,  y.  The  prime  factor  2  occurs 
once  in  the  second  expression,  hence  it  occurs  once  in  the  L.  C.  M. 
Similarly,  the  prime  factors  3  and  5  occur  once  in  the  L.  C.  M.  The 
prime  factor  a  occurs  once  in  the  second  expression,  but  twice  in  the 
first  expression  ;  hence  it  will  occur  twice  in  the  L.  C.  M.  Similarly, 
the  prime  factor  x  occurs  twice  in  the  L.  C.  M,  Finally,  the  prime 
factory  occurs  once  in  the  L.  C.  M.  Collecting  results,  we  see  that 
the  L.  C.  M.  is  equal  to  2X3X5  ««  ^  ^/,  or  30a^x~y. 

Find  the  L.  C.  M.  of  the  following  expressions: 

1.  Ux'^y'^  and42xy^z.  4.   17a'^t?'^c-  and  17 a^d^c^. 

2.  7xjyz  and  Sax'^y^z'^.  5.   27x'^y^2^  and  2m'^x^y. 

3.  dadc  and  15a'^d^x^y.  6.   14x'^y'^, 9adc,  and  7  xys. 

7.  42xy^2:,    ^ax'^y^z^,  and  bw^. 

8.  a'^b'^c'^x^y^z^ ,    abcu^v^w"^ ,  and  uvwxyzabc. 
9.  hax,     lOay,     25b'^z^,     lOOa'^c^,  and  50abcxy^z. 

10.  brst,     12,     rt,     30,     20r/2,     Ibst^ ,  and  4s^t. 

11.  14rV6,     2Uc\     70a''bc,     10^oad\  and  ZUcd'^ . 

12.  2bcd\     15ad^,     14b^c^,    lOb^d^    21  a'' c^,  and  S5bd. 

13.  5ac\     lOac',     7d\     ^a'^b'',     lb b^c^,  and  14ad'^ . 

14.  30«2^V3,  70ac''d\  42a''b^d\  l^hb^c'^d^ ,  and  Zhc^dK 

15.  20jr|/2^,  4hxz'^v'',  ZQzu'^v^,  Sxyzu'^v^,  and  5  u^v^, 

16.  QOzv,     45xz^v,     dOy'^zuv,     9xz7c'^v,  and  30;t:z;^. 


FRACTIONS.  Ill 

17.  Multiply  the  L.  C.  M.  of  Sa-xy  and  2ax^j'^  by  the 
H.  C.  F.  of  the  same  expressions. 

i8.  Divide  the  product  o{  a-dx  and  ad  by  the  H.  C.  F. 
of  a'^dx  and  ad,  and  compare  the  quotient  with  the  ly.  C. 
M.  of  a-dx  and  ad. 

19.  Divide  the  L,.  C.  M.  of  2a'^x^y  and  3  ax'^jy  by  the 
first  of  the  two  expression,  and  compare  the  result  with 
quotient  obtained  by  dividing  the  second  of  the  two  ex- 
pressions by  the  H.  C.  F.  of  the  two  expressions. 

20.  Divide  the  product  of  ba^x^j^  and  15adx*2,  by  the 
L.  C.  M.  of  the  same  two  expressions,  and  compare  the 
quotient  with  the  H.  C.  F.  of  the  same  two  expresssions. 

EXERCISE  58. 

Fractions. 

88.  We  have  already  used  the  fractional  form  j  as 

another  way  of  writing  a-i-d,  so  that  -r  is  an  expression 

of  division.  We  have  already  learned  that  in  any  case 
of  division  the  divisor  multiplied  by  the  quotient  equals 
the  dividend,  or,  in  the  language  of  fractions,  precisely 
the  same  thing  may  be  written, 

Denominator  x  Quotient  =  Numerator. 

89.  From  this  equation  it  is  plain  to  see  that  if  the 
denominator  remains  unchanged,  multiplying  the  numer- 
ator by  any  number  multiplies  the  quotient  by  the  same 
number,  and  dividing  the  numerator  by  any  number 
divides  the  quotient  by  the  same  number,  or,  as  it  is 
more  often  stated,  multiplying  the  numerator  dy  any  num- 
der  multiplies  the  fraction  dy  that  numder,  and  dividing  the 
numerator  dy  any  7iumder  divides  the  fractioji  dy  that 
numder. 


112  FACTORS,   MULTIPLES,   AND    FRACTIONS. 

90.  Again,  from  the  same  equation, 

Denominator  X  Quotient  =  Numerator, 
it  is  also  plain  that  if  the  numerator  remains  unchanged, 
multiplying  the  denominator  by  any  number  divides  the 
quotient  by  that  number,  and  dividing  the  denominator 
by  any  number  multiplies  the  quotient  by  that  number, 
or,  as  it  is  more  often  stated,  mtiltiplying  the  denominato? 
by  any  number  divides  the  fraction  by  that  number,  and 
dividing  the  deiiominator  by  any  number  multiplies  the 
fraction  by  that  number. 

91.  Once  more,  from  the  same  equation. 

Denominator  X  Quotient  =  Numerator, 
it  is  plain  that  if  the  quotient  remain  unchanged,  multi- 
plying the  denominator  by  any  number  multiplies  the 
numerator  by  the  same  number,  or,  stated  in  another 
way,  multiplying  both  numerator  and  denominator  by  the 
same  number  does  7iot  alter  the  value  of  the  fraction.  It  is 
also  evident  from  the  same  equation  that  dividing  both 
numerator  and  denominator  by  the  same  number  leaves 
the  quotient  unchanged,  or,  stated  in  the  usual  form, 
dividing  both  numerator  and  denominator  by  the  same  num- 
ber does  not  alter  the  value  of  the  fraction. 

92.  When  the  numerator  and  denominator  of  a  frac- 
tion are  each  divided  by  all  the  factors  common  to  both 
numerator  and  denominator  so  that  the  resulting  numer- 
ator and  denominator  contain  no  common  factor,  the 
fraction  is  then  said  to  be  in  its  Lowest  Terms. 

Of  course,  then,  to  reduce  a  fraction  to  its  lowest  terms 
we  divide  both  numerator  and  denominator  by  every  factor 
common  to  both,  i.  e.,  by  the  H.  C.  F.  of  the  numerator  and 
denominato? . 


ADDITION    OF    FRACTIONS.  "    II3 

Reduce  the  following  fractions  to  their  lowest  terms  : 

7.  T 


I. 

oax 

\Ux'^' 

2. 

32^2  j«:2 

^'^bcx  ' 

Vlabc 

^       ZSixz^  '     187aiijir9j/5- 
25wV*£^  209r3^V*z; 

Ibcde'  '  liy^axy  ^'    SOlaH^z' 

S00x*y^zu^ 


EXERCISE  58. 

Addition  of  Fractions. 

93.  If  two  or  more  fractions  have  the  same  denom- 
inator, the  fractions  may  be  added  by  adding  the 
numerators  and  placing  the  sum  over  this  common  de- 
nominator; but  if  the  denominators  are  not  the  same  we 
must  multiply  the  numerator  and  denominator  of  each 
fraction  by  such  a  number  as  will  make  all  the  denom- 
inators the  same,  and  ^/ie?i  add  the  numerators,  and  place 
the  sum  over  this  common  denominator. 

94.  The  process  of  changing  the  numerators  and  de- 
nominators of  fractions,  so  that  each  fraction  shall  pre- 
serve the  same  value  it  had  before,  while  the  denominators 
are  all  made  alike  is  called  Reducing  to  a  Common  De- 
nominator. 

For  example,  suppose  we  wish  to  add  — -  and  —  to- 

m^  mn 

gether.     We    must   reduce  to  a   common   denominator, 

which,   of  course,   must  be  a  common  multiple   of   w- 

and  mn.     Any  common  multiple  will  do,  but  the  lowest 

common  multiple  is  preferable. 


114         FACTORS,   MULTIPLES,   AND    FRACTIONS. 

The  lowest  common  multiple  is  plainly  vi'^n\  hence,  we 
must  multiply  the  numerator  and  denominator  of  the 
first  fraction  by  ?2,  and  the  numerator  and  denominator  of 
the  second  fraction  by  m.     We  then  have 

Plainly,  then,  the  sum  of  the  two  fractions  is 
Aiabn-\-^mx 
m'^71 
Hence,  we  may  write  the  equation, 
ab      4:X  _  4ab?i  +  4mx 
m'^     7nn  m'^71. 

If  we  had  three  fractions  to  add  together,  would  the 
common  denominator  be  a  common  multiple  of  each  of 
the  given  denominators  ?  Would  there  be  any  preference 
for  the  lowest  common  denominator  ?     Why  ? 

If  any  mmiber  of  fractions   are  to  be  added  together, 
what  would  you  prefer  to  take  for  a  common  denominator  ? 
State  the  method  to  be  pursued  when  two  or  more  frac- 
tions are  to  be  added  together. 
Add  the  following  fractions  : 

2«       ,  3«  cd         ^     cd  m^  r'^       ,    r'^s'^ 

and  7y-.      3.  --^.  and  -— x.   5.  -^— -  and 


X           2x'        '  x'^ y  xy^'      '     bxy           Ibx 

ab       ^      X            abc  ,    ab^^c^    _     Aiab        .    4cd 

2.  —  and  — T— .  4. and  - — —-:^.  o.  ^^r— ,  and  t^—,. 

m           ni^n         xyz  xyz""         Ilea          liao 

Ub''       ^     Iz  lOa'-b^       ^    7 

_      \Qrst        ,    IQrst  12uv       ,   13^/ 

8.  — and  -^ .  10.  -tr^--  and  -.-^ — . 

a        X         y"^  Auv                  12 

x"^      m^      u  14xy         ,     7 

12.  — ^,     — ^,     -  — ^-,  and  -^- 


3* 


SUBTRACTION    OF    FRACTIONS.  II  5 

EXERCISE  59. 

Subtraction  of  Fractions. 

If  one  fraction  is  to  be  subtracted  from  another  and  the 
fractions  have  the  same  denominator,  we  may  subtract 
the  second  numerator  from  the  first  and  place  the  re- 
mainder over  this  common  denominator ;  but  if  the 
denominators  are  different,  we  must  first  reduce  the 
fractions  to  a  common  denominator  and  then  perform  the 
subtraction. 

^         Sx     ,      Ax  ^         \babc  ^  ,       xyz 

1.  From  TT- take  5— I.        3.  From take  .,  r    ,  . 

2a  3^2  ^  xyz  \habc 

Za'^x     ,     4cx  ^         lOaHc.       Vlab 

2.  From  —^rr-  take  ^r^-     4-  From  take 


2b  3^2-     t- ;^5^^^   ""^^^20^2^ 

5.  From— p— take  ^2_^2- 

(xbc^  cdc"^ 

6.  From  y-j^  take  -7-7^. 

bcd^  dej  ^ 

7.  From  -^-  take  j^-. 

12  i  xyz 

„    ^  '?^ab        ,        V2xz 

o.   From  jr r,  take  771 — ^-. 

\)xyz~  Ibxz-u 

^           Ma        .         Qx 
9.  From  fTz take  7^^ — ^— r. 

12mr^      .      14;z^2 


10-  I^rom  ^^^^:— y-  take  -, 


20;«V  2l7i's 


Il6  FACTORS,   MULTIPLES,  AND    FRACTIONS. 


EXERCISE  60. 

Multiplication  of  Fractions. 

95.  We  are  to  multiply  -r  by  -7. 

Now  to  multiply  by  -3  means  to  multiply  by  c,  and  di- 
vide the  result  by  d.     Therefore  to  multiply  t  by  -  we 

first  multiply  t  by  c,  and  then  divide  the  result  by  d. 

We  may  multiply  a  fraction  by  multiplying  the  nu- 

,  a  ac 

merator;  hence,  -7X^=-r. 

o  0 

We  may  divide  a  fraction  by  multiplying  the  denom- 

.     ,        -  ac      .    ac 

inator;  hence,  -r-i-«=T3- 

0  oa 

/TM       r  a     c     ac 

Therefore,  -7  X  -7 = -r> 

0    a     oa 

Hence  to  multiply  two  fractions  together  multiply  the 
numerators  together  for  a  new  numerator^  and  the  denom- 
inators together  for  a  7iew  demoninator. 

96.  Suppose  we  wish  the  product  of  three  fractions, 

r     ■     ^  ace 

as  for  instance,  -7  x  -7  x  >. 

0    d    f 

We  may  multiply  the  first  two  together,  as  just  ex- 
plained, giving  ^ 

and  we  can  then  multiply  this  result  by  the  third  fraction, 
by  the  method  just  explained,  giving 

ace 

'bdf' 

ATvt,      c  ace     ace 

Therefore,  x  -  X  -r= 7-^ . 

b    d   f     bdf 


MULTIPLICATION   OF    FRACTIONS.  II7 

As  this  can  be  extended  to  any  number  of  fractions, 
we  have  : 

The  product  of  any  nuviber  of  fractio7is  is  found  by  mul- 
tiplying all  the  numerators  together  for  a  new  numerator, 
and  all  the  de7iominators  together  for  a  new  denomhiator. 

Multiply  the  following  fractions  : 

^ab        .    5fd  a^x"^  bxy         ,   Qbys 

'•  A^i  ^"'^  6i^-  3.  -j^,,     --.-.  and  ^. 

4xy       ,       Suvw                7ir          u^  x         ,     v^ 
2.  7;-^  and  — —T— .     4. , 1,  and  — r. 

b  7iv  Vlxyz  21V  wx*  xz^ 

Arstu'^v        lOx'^y  labu'^v'^ 

buvxy '      \^u^v^-t'  rsx 

143  nrs      Six^z  2w^  bab^      Icd^  %e£ 

'  ISlxyz'      '6\)wt'  Ss  '    ^'  6xy     Siw'  ^^^  70w 

a^        a^         a^  a^ 

x^'       x^'       x^'  x^' 

^2^2^2       2  ^    7ia  a         b        bx         ^  y^ 

9.  — :j — , ,  and -irr-     ^°-  ""' » >  and -^. 

1         7n'w  a- be  x        y        ay  b^ 

a*        x'^        y'^z  ,       w'^xyz 

II. i, ,     ■^, — ^,  and ~-, 

X*         y        z-w^  abc 

abc      a^x         bz^  cy 

12.  ,     -ry-, --,  and 5—3. 

xyz       b^z        xy^  xy^z^ 

2rst^       42jr^2       3^,^^,^  l^r'^x 

^3.  1---1.       1 A ..  >      o^.-  >  and 


4xyz^'       Uy-^'      27yz'  SSrxz^' 

571VW         187ivz^  Vl7ivxy 

14-    ^r~i 1   — ^  -■   "  o   >   and  -r:: r. 

Vlxyz        lovw^y  Sovwz* 

Idabrst^  51az>s'^x  IQawx^z 

\l7ivn^xy       64:b7(r'^wy'  d7ivyz^ 

m'^n^r^s^       bux^fy"^  ,        m^r^ 

16. ,     — = — f^,  and TT — . 

xyz      '         Icz^    '  9r 


Il8        FACTORS,   MULTIPLES,   AND    FRACTIONS. 


i8. 


axys        d'^ti'^x'^       w^x         .  tvw 

7     -' ^^-^'     — 1->   a^^  ■ 

Duvw         r-y^z-'       yz*  xyz 

abk        mr        r'^kv         ,  l(Smii 

19.  ■ ^,     -^,   —  - — ^,  and  -p . 

uvx^        u^         wx"^  bxyz 

ax       b'^z        A.WX         .  rxy 

by       c^y        mz-  syz 


EXERCISE  61. 

Division  of  Fractions. 

97.   We  are  to  divide  -.-  by  -3. 
0        a 

We  may  write  the  quotient  in  the  form  of  a  fraction, 

where  the  numerator  is  itself  the  fraction  -r  and  the  de- 

0 

nominator  is  the  fraction  — . 

a 

a 
Hence  —=  quotient. 

Let  us  multiply  both  numerator  and  denominator  of  this 
fraction  by  bd.  We  know  this  will  not  change  the  value 
of  the  quotient. 


Hence 
Therefore 


a 

1'' 

bd= 

abd 
''   b~~ 

=ad, 

c 

bd= 

bed 
d 

--be. 

quotient= 

ad 
''be- 

a 
T 

c 

ad 
~  be' 

DIVISION    OF    FRACTIONS.  II9 

This  result  ma}^  be  obtained  by  multiplying  the  fraction 

-.  by  the  fraction  — ,  which  last  is  the  divisor  inverted. 
0  c 

Hence,  to  divide  one  fraction  by  another,  invert  the 

terms  of  the  divisor  and  multiply. 


2.1-        \z  ^.   .^        Imr  ^     ^r'^t 

,,-  by  -^~.  3.  Divide  —7.7-  by  -^-^. 

6v        bv  btx         Ix^ 


_.    .,      rst   .          2nv  _.    .,    34jt-i'         2y^z 

2.  Divide  — —  by  — ^ — .  4.   Divide    ^^        by  -^ — r, 

uvw            6sw  oliiv          6vw^ 

^.    .^         \Omk''  ^  ZUs^ 

5.  Divide -^r-j-  by  —  ,,^    ^  . 

b;'5^  blx-2 

6.  Divide  ^,—  bv 


5to     -   lOk'-xz 

<S  3r  6;»;2 

7.   Divide  the  sum  of  ~  and  -^  by  the  sum  of  — ^ 


and  — ^. 

8.   Divide  the  sum  of  — ,,-  and v  by  the  product  of 

-  and  —7. 
.;f  x^ 

x^         7iZ'  ffi''n 

0.   Divide  --^  by  -.,    ,  and  divide  the  result  by  — r— . 

10.  Divide  ^  by ,  and  multiply  the  result  by  j . 

a'"^         7'st  uvx^ 

11.  Divide  —  by ,  and  multiply  the  quotient  by --^^-^ 

X         xyz  ^  a"y^ 

7?i  rt  1 

12.  Divide  -:7  by  — ;,  and  divide  the  quotient  b}^ ^— 

s^        uv  r-st 

13.  Divide  -7  by -7,  and  multiply  the  quotient  by 

21      s 
the  quotient  of — '--. 


I20        FACTORS,   MULTIPLES,   AND    FRACTIONS. 


14.  Multiply  mr'^x'^  by  the  quotient  of  — ^h . 

15.  Add  the  quotient  of — =^h — ^--  to  the  quotient  of 
axw     uvx 


rty       sxt 

y's"^'  X  tixTJU 

1 6.  Add  the  product  of  -^ ^  and  —z—  to  the  quotient 


of 


luvy  '    2vy^ 


Tst       tnvx^ 

17.  Subtract  the  quotient  of '--- — —from  the  prod- 

uvw     vw-y 

,  uvx       ,    r^^y 
uct  of  —r-  and  -~-, 
aoc  u^xz 

18.  Divide  —1^—  by  -^— ,  anH  divide  the  result  by 

x^y         rsx  -^       axz 

fSX^ 

and  divide  ^hts  result  by  —7, — . 

u-vy 

CL^  X  hv 

10.  Divide    -tt, —  by  3,  divide  the  result  by  -r—,  and 
b-yz  AlX^ 

divide  this  result  by  the  product  of  — ^  and . 

x-^  zvy 

1           abx"^ 
20.  Multiply  — —  by ^  and  divide  the  product  by 

rsty        uvy^ 

the  product  of  — ^-  and  —Trr- 
^  x^y         a^bz 


CHAPTER  X. 

SIMPLE  EQUATIONS. 

EXERCISE  62 

Definitions  and  General  Principles. 

98.  An  Equation  is  the  statement  of  equality  which 
exists  between  two  expressions. 

99.  The  Members  or  Sides  of  an  equation  are  the 
parts  on  either  side  of  the  sign  =,  and  are  distinguished 
as  the  First  and  Second  Members,  or  Left  and  Right 
Sides,  respectively. 

Students  often  have  a  careless  habit  of  calling  almost  everything 
in  Algebra  an  equation.  Thus,  v^^e  hear  a'^-\-1ab-\-b'^  called  an  equation 
instead  of  an  expression  or  a  trinomial.  It  is  better  to  call  an  expres- 
sion an  expression,  and  an  equation  an  equation.  It  is  also  better  to 
say  "multiply  both  members  oi  the  equation  by  2,"  than  "multiply 
the  equation  by  2,"  etc. 

100-  If  the  two  sides  of  an  equation  are  equal,  no 
matter  what  numbers  be  substituted  for  the  letters,  the 
equation  is  called  an  Identical  Equation  or  simply  an 
Identity.     Thus,  the  following  equations  are  identities: 

x(x—a)=^x^—ax, 
{x-\-a)(^x—a)=x-—a^, 
for  the  equations  are  true,  no  matter  what  numbers  be 
put  for  X  and  a. 

101.    If  the  two  sides  of  an  equation  are  equal  only 
when  a  particular  number  or  numbers  are  substituted  for 


122  SIMPLE    EQUATIONS. 

the  letters,  the  equation  is  called  a  Conditional  Equa- 
tion or  simply  an  equation.  The  following  are  condi- 
tional equations  :  ;i:-f  1  =  2, 

for  the  first  equation  is  only  true  when  x=l  and  the 
second  is  only  true  when  x=S. 

102.  A  letter  for  which  a  particular  number  must  be 
substituted,  in  order  that  the  two  sides  of  the  equation 
may  be  equal,  is  called  an  Unknown  Number. 

103.  A  number  which,  when  substituted  for  the  un- 
known number,  makes  the  two  sides  of  the  equation 
equal,  is  said  to  Satisfy  the  equation,  and  is  called  the 
Root  of  the  Equation  or  a  Value  of  the  Unknown 
Number. 

104.  To  Solve  an  equation  is  to  find  the  root  or  roots. 

106.  A  Simple  Equation  or  Equation  of  the  First 
Degree,  is 'one  which  contains  only  the ^rs^ power  oi  the 
unknown  number.     Thus, 

8 — }rx — 4.r  =  — - — 
4 

is  a  simple  equation,   but   x''--\-Ax=b    is    not  a  simple 

equation. 

106.  The  axioms  which  are  useful  in  the  solution  of 
simple  equations  are  as  follows  : 

/.  If  we  add  to  equals  ihe  same  number  or  equal  num- 
be7s,  the  sums  will  be  equal. 

II.  If  we  lake  from  equals  the  same  tutmber  or  equal 
numbers,  the  remaiiiders  will  be  eqiLal. 

Ill  If  we  multiply  equals  by  the  same  number  or  equal 
numbers,  the  products  will  be  equal. 

IV.  If  we  divide  equals  by  the  same  7iumber  or  equal 
numbers,  the  quotients  will  be  equal. 


DEFINITIONS   AND    GENERAL    PRINCIPLES.       1 23 

107.  The  first  two  of  these  axioms  are  embodied  in 
the  Principle  of  Transposition  :  A7iy  term  in  one 
7?ie7nber  of  an  equation  may  be  transposed  to  the  other  mem- 
ber^ provided  its  sig7i  be  changed. 

108.  The  use  of  the  axioms  is  illustrated  by  the  fol- 
lowing examples  : 

X 

(1)  Solve  the  equation  -p=6. 

o 

Multiplying  both  sides  of  the  equation  (axiom  3)  by 
5,  we  get  jr=30. 

Zx 

(2)  Solve  the  equation  -77- =  9. 

Multiplying  both  sides  of  the  equation  (axiom  3)  by  7, 
we  have  3.r=G3. 

Dividing  both  sides  of  the  equation  (axiom  4)  by  3,  we 
obtain  ^=21. 

(3)  Solve  l+|+|=;»:-7. 

Multiplying  both  sides  of  the  equation  (axiom  3)  by  6, 
we  have  6  +  3ji:+2;»;=6.r— 42. 

Transposing  (axioms  1  and  2)  unknown  numbers  to  the 
left  side  and  known  numbers  to  the  right  side,  we  get 

3jr+2j»;— 6jr=— 42— 6. 
Uniting  similar  terms,        — .r=— 48. 
Dividing  both  sides  (axiom  4)  by  —1,  we  have 

.;i:=48. 

(4)  Solve  f-|+f =2-1 +ff. 

Multiplying  both  sides  of  the  equation  (axiom  3)  by 
12,  we  have      6x—4x-\-Sx=24:—2x+5x. 
Transposing  unknown  numbers  to  left  side, 

iJ.\-—4x-\-Sx-\-2x—ox=24:. 
Uniting  similar  terms,     2x=24:. 
Dividing  both  sides  (axiom  4)  by  2,  ;r=12. 


124  SIMPLE    EQUATIONS. 

109.  The  object  in  multiplying  both  sides  of  an  equa- 
tion by  the  same  number  is  to  Clear  the  Equation  oi 
Fractions.  This  may  be  accomplished  in  two  ways  : 
First,  multiply  both  members  of  the  equation  by  the  product 
of  the  deno77iinators  of  all  the  fractions. 

Or,  if  we  prefer,  we  may  multiply  both  members  of  the 
equation  by  the  least  com^non  de7iomi7iator  of  all  the 
fractions. 

Thus,  solve  the  equation  ^-  +  ^+^-306=0. 

Multiplying  both  sides  by  60,  the  least  common  denom- 
inator of  the  fractions,  we  get 

45;r+28;r+110^-21960=0. 
Transposing  known  quantity  to  right  side  and  uniting 
similar  terms,  we  have     1 83;*;=  21960. 
Dividing  both  members  by  183,  we  obtain 
;r=120. 

110.  In  clearing  an  equation  of  fractions  the  student 
must  take  special  care  when  a  minus  sign  occurs  before 
a  fraction  and  the  numerator  is  not  a  monomial.  It 
must  be  remembered  that  the  dividi7ig  li7ie  of  a  fractio7i  is 
the  same  as  a  parenthesis  enclosi7ig  the  7iume7^ator.     Thus, 

^_l_^  jf-f-4 

2 — is  the  same  as  2— (;r+4),  and  2 ^—  is  the 

1  o 

is  the  same  as  2— 4(jt--f  4).     We  will  illustrate  this  point 
by  a  few  examples. 

(1)  Solve -|^=-^ ^+1^- 

Multiply  both  sides  by  12, 

4(jt:+2)  =  3(5-;»:)-6(;t:4-l)  +  168. 
Perform  the  indicated  operations, 

(4;»;+8)=(15-3;r)-(6;t:+6)-fl68. 
Remove  parentheses, 

4.r+8=15-3;r-6;»;-6  +  168. 


DEFINITIONS    AND    GENERAL    PRINCIPLES.       I25 

Transposing  the  known  numbers  to  the  right  side  and 
the  unknown  numbers  to  the  left  side, 

4;»:  +  3jr+6^=15-6  +  168-8. 
Uniting  similar  terms, 

13;«;=169; 
whence  jt=13. 

(2)  Solve;*: ^—=3 ^— . 

O  o 

Multiply  both  sides  by  15, 

15;<;-3(3;»;-2)  =  45--5(2;»;-5). 
Perform  indicated  operations, 

•     15;i:-(9;i:~6)=45-(10.r-25). 
Remove  parentheses, 

15Ar-9j»r+6=45-10;»;-f25. 
Transpose  the  known  numbers  to  the  right  side  and  the 
unknown  numbers  to  the  left  side, 

15;»;-9j»;+10ji:=45  +  25-6. 
Uniting  similar  terms,     16;r=64  ; 
whence  x=4. 

111.  Tke  signs  of  all  the  terms  in  the  mwierator  of  each 
fractio7t  preceded  by  the  juinus  sigji  must  be  chaiiged  when 
the  equation  is  cleared  of  fractions.  The  neglect  of  this  is 
a  very  commo7i  source  of  error. 

112.  We  will  now  formulate  the  Method  for  Solving 
any  Simple  Equation. 

/.      Clear  the  equation  of  fractions. 

II.  Transpose  the  knoivn  ^lumbers  to  the  right  side  a7id 
the  unknown  numbers  to  the  left  side  of  the  equatioii. 

III.  Unite  similar  terms. 

IV.  Divide  both  sides  by  the  coefficie^it  of  the  unknown 
quantity. 

This  is  generally  the  best  order  to  pursue,  although  it  is  sometimes 
shorter  to  unite  similar  terms  before  clearing  of  fractions. 


126  SIMPLE    EQUATIONS. 


EXERCISE  63. 

Examples, 

Solve  the  following  equations  : 

I.  1+8=13. 

X         A:X          ^ 

9-2,9  =^- 

5      3 

^"  6     4x' 

'°-  2  +  3-4=7- 

3.  ^^5=8. 

X 

xz.  f +-|=9. 

XX       - 

4.  2  +  3=5. 

-.  r+^^=22.' 

X       1       X 

^'  3'^6~2- 

Zx     2x     ,„ 
^3.    4-7=13. 

6.  f +7=.-f. 

^4-  f +T-f =9- 

7.  3  +  5=8. 

8:ir   ,    7x 

15.   1.3;^— g-+^= 

8.  1^+1^=38. 

2^     3^     ^ 

15jt:  +  22 


3       3.r     1     2jc  .  ^ 

17.  2^-5^'=^-2-^+2. 

-»  J^  J-V  ^  ^  J4^  ^  ,    J 

'«•  2-3+4-6+8  +  r2=^^- 
In  the  following  it  will  secure  greater  accuracy  if  the 
•student  will  put  the  work  down  in  full,  as  in  Art.  110. 

'Thus,  in  the  next  example  the  fraction ^ —  must  be 

multiplied  by  6,  and  it  is  better  to  write  the  result  in  the 
form  —  2  (5  jr  +  4 )  first,  and  afterwards  in  the  form 
— 10,^—8,  in  order  to  avoid  7?iistakes  i7i  the  signs. 


LITERAL    EQUATIONS.  12/ 

X     hx-\-\     4jr— 9 

x-1     a-+23      10  +  ^ 

20.    X — 


o 


4  5 


3j»:+9     5jf+16  5;t-— 6     Zx     x—4 

21.     ; = ^ .  23.     z T^  =  — Q—- 

4  /  o  lo         9 

5-ji:     n-Sx  12-3a-     3.r-ll     , 

2(7jf-10)     ,^^     50-jtr 
25.   -^--^-20=-^. 

4  o 

„    5x— 6     3^—4     4;tr    ^ 

28.  ^ 0— ri=o- 


-^y- 

2 

1 

6     ' 

8 

8     • 

30. 

5;»:- 

11 

jf-i 

11;«:- 

-1 

4 

10  ■ 

~      12 

* 

31. 

8-;»: 

1 

3.r-5 

x-h6 

X 

6 

1 

3 

3' 

Jt:  +  3       1  o^       1  /Q         -N   ,   1 

32.   — ^ ^(;i:-2)  =  -^(3^-o)+^. 


EXERCISE  64. 

Literal  Equations. 

113.  If  the  known  numbers  in  an  equation  are  repre- 
sented by  letters  instead  of  by  figures,  the  equation  is 
spoken  of  as  a  Literal  Equation.  Of  course  no  differ- 
ent principles  hold  in  the  solution  of  such  equations,  than 

in  the  solution  of  numerical  equations. 


128  SIMPLE    EQUATIONS. 

It  must  be  remembered  that  the  first  and  intermediate 
letters  of  the  alphabet  stand  for  numbers  supposed  to  be 
known  or  given. 

Solve  the  following  literal  equations: 

I.  6fnx-{-2a=8mx—2d. 

Transposing  the  unknown  numbers  to  the  left  side,  and  the  known 
numbers  to  the  right  side,  we  obtain 

6mx—8mx=—2i  —  2a.. 
Uniting  similar  terms 

—  2mx=—2l>—2a. 
Dividing  both  sides  by  —2/;/ 

_a-|-3 
~    m  ' 
which  is  the  value  of  x. 

ax—b     bx-\-c 

2. =aoc. 

c  a 

Multiply  both  sides  by  ac 

a[ax  —  h')  —  c[l)x-\-c^^=^a.^bc*. 

Remove  the  parentheses 

a^x—ab—bcx — c^r=za'^bc*. 

Transposing  known  numbers  to  right  side 

a^x  —  bcx^=^a'^  bc'^  -\-ab-\-c^ , 

Uniting  with  a  parenthesis 

{a^—bc)x—a^bc^^ab^c*. 

Dividing  both  sides  by  {a^—bc) 

■    _a^bc^-\-ab-\-c^. 

^~         a^-bc        ' 

3.  x-\-a=d,  8.  77ix=r. 

4.  Sx-\-6n==dr.  9.  mx-\-p—r. 

5.  a-\-b—x—^.  10.  V8-\-ax—b=2x. 

6.  x-\-a—b—c.  II.  x-\-hx—b==2a. 

7.  J)/— 3  +  ^=8.  12.  Za-^2y—U=-hy—b. 

13.  6a-8^+6«j/=4«-6^-2^+8m'. 

14.  hmx—^a—\\b=Z7nx—^b-\-lc—hmx—2a. 


SYMBOLIC   EXPRESSIONS.  1 29 

15.  12(x-c)  =  7(ic+x).  20.  5(5c-2x')  =  4x-Qc. 

16.  S(a—5x)=4:(Sx—2c).  21.  a(bx—c)=ac—adx. 

17.  7(a—x)=6(d—x).  22.  ;^;r— r=;ir. 

18.  (d—l)j=d—j/.  23.  ad—dy=-my—am. 

19.  (l  —  /^)ji:=  «—;»;.  24.  /)(;t:— l)-f  •^=^— /. 

25.  r(l4-jr)  +  «(r+/>)=r;i:+w/+;ir. 

26.  «r— ;?(_>/+ 1)+j=?2(2—jk). 

27.  (^+l)j»;+a/^=^(«+^)  +  «. 

„    U{^x-a)  ,  (jt:-^2)         (4a-\-cx)d 

20.    p 1 r^r-7 = ^ . 

ba  loo  oa 

nx     r—x  ,  7i(r—x) 
^9.  V— 2^-+-3^  =  "- 


30. 


"lax— 2b     ax—a_ax      2 
3^^  2b~~~b  ~  3* 


^4-<:  ,         a     ,^  a{a-V)-b{b-V)^-c 

^  X^  X         ^  X 


EXERCISE  64a. 

Symbolic  Expressions. 

1 14.  In  solving  a  problem  in  Algebra  we  must  not  only 
select  some  letter  to  stand  for  the  unknown  number,  but 
we  are  required  to  find  expressions  which  will  sj'^mbolize 
all  other  numbers  which  occur  in  the  problem.  Such 
may  be  called  Symbolic  Expressions.  Thus,  if  the 
sum  of  two  numbers  is  100,  and  if  x  stands  for  one  of 
them,  then  the  expression  100— ji:  stands  for  the  other 
number.  If  x  is  the  price  of  one  horse,  then  V^x  stands 
for  the  cost  of  10  horses;  if  5  yards  of  cloth  cost  x  dol- 
lars, then  the  cost  of  one  yard  is  represented  by  the  ex- 

X 

pression  -,  etc.     Some  drill  in  the  formation  of  symbolic 
expressions  will  be  of  help  in  the  solution  of  problems. 


130  SIMPLE   EQUATIONS. 

1.  The  sum  of  two  numbers  is  85.  The  first  number 
is  8,  what  is  the  second  ?  The  first  number  is  n,  what  is 
the  second  ? 

2.  The  sum  of  two  numbers  is  a.  The  first  number  is 
5,  what  is  the  second  ?  The  first  number  is  b,  what  is 
the  second  ? 

3.  A  train  travels  at  the  rate  of  20  miles  per  hour  for 
3  hours;  how  far  does  it  go?  A  train  travels  at  the  rate 
of  r  miles  per  hour  for  3  hours;  how  far  does  it  go  ?  A 
train  travels  at  the  rate  of  r  miles  per  hour  for  /  hours; 
how  far  does  it  go  ? 

4.  What  must  be  added  to  100  to  make  a  ?  What  must 
be  added  to  x  to  make  a  ? 

5.  One  factor  of  100  is  10;  what  is  the  other?  One 
factor  of  100  is  x\  what  is  the  oth^? 

6.  What  two  numbers  differ  from  100  by  7  ?  What  two 
numbers  differ  from  100  by  ^  ?  What  two  numbers  differ 
from  nhy  xt 

7.  How  much  will  n  apples  cost  at  c  cents  apiece  ? 

8.  If  one  apple  costs  2  cents,  how  many  can  you  get 
for  X  cents  ?  If  one  apple  costs  c  cents,  how  many  can 
you  get  for  x  cents?  How  many  can  you  get  for  d 
dollars  ? 

9.  How  many  hours  will  it  take  to  go  x  miles  at  4 
miles  per  hour  ?  How  many  hours  will  it  take  to  go  x 
miles  at  r  miles  per  hour  ?  How  many  minutes  will  it 
take? 

10.  A  train  goes  150  miles  in  h  hours;  what  is  the  rate 
per  hour  of  the  train  ?  A  train  goes  m  miles  in  h  hours; 
what  is  the  rate  of  the  train  ? 

11.  The  rate  of  a  train  is  r.  How  far  will  it  go  in 
time  /  ? 


SYMBOLIC    EXPRESSIONS.  131 

12.  What  is  the  interest  on  d  dollars  for  t  years,  at  5 
per  cent.?  What  does  the  principal  and  interest  amount 
to  for  this  time  ?  What  is  the  interest  on  d  dollars  for  / 
years  at  r  per  cent.? 

13.  A  man  can  do  a  piece  of  work  in  10  days.  How 
much  of  it  can  he  do  in  one  day?  A  man  can  do  a 
piece  of  work  in  n  days.  How  much  of  it  can  he  do  in 
one  day  ? 

14.  A  can  do  a  piece  of  work  in  9  days  and  B  can  do  it 
in  12  days.  What  part  can  each  do  in  one  day  ?  What 
part  can  both,  working  together,  accomplish  in  one  day  ? 
A  can  do  a  piece  of  work  in  a  days  and  B  can  do  it  in  b 
days.  What  part  can  both,  working  together,  accom- 
plish in  one  day  ? 

15.  A  pipe  will  fill  a  cistern  in  7  hours;  what  part  runs 
in  during  one  hour  ?  Another  pipe  will  fill  the  cistern 
in  5  hours;  what  part  runs  in  during  one  hour?  The 
cistern  holds  x  gallons.  How  many  gallons  does  each 
pipe  carry  in  one  hour  ? 

16.  One  pipe  will  fill  a  cistern  in  a  hours,  and  another 
pipe  will  fill  it  in  b  hours.  What  part  does  each  pipe 
carry  in  one  hour  ?  What  part  do  both  pipes  together 
carry  in  one  hour  ? 

17.  The  digit  5  stands  in  tens'  place;  what  number  is 
expressed?  The  digit,  represented  by  x,  stands  in  tens' 
place;  what  number  is  expressed  ?  A  digit  represented  by 
X  stands  in  hundreds'  place;  what  number  is  expressed? 

18.  If  the  first  and  second  digits  of  a  number  are  5 
and  7  respectively,  what  is  the  number?  If  the  first  and 
second  digits  of  a  number  are  represented  by  a  and  a +  2 
respectively,  what  is  the  number  ? 


132  SIMPLE   EQUATIONS. 

19.  The  three  digits  of  a  number  beginning  at  the 
right  are  represented  by  .r,  x+2,  and  x—S;  what  is  the 
number  expressed  by  them  ? 

20.  Write  three  consecutive  numbers  of  which  7  is  the 
first.  Write  three  consecutive  numbers  of  which  n  is  the 
first. 

21.  Write  three  consecutive  even  numbers  of  which  6 
is  the  first.  Write  three  consecutive  even  numbers  of 
which  2n  is  the  first. 

22.  Write  three  consecutive  odd  numbers  of  which  5 
is  the  first.  Write  three  consecutive  odd  numbers  of 
which  2«  +  l  is  the  first. 

23.  Write  three  consecutive  numbers  of  which  n  is  the 
greatest.  Write  three  consecutive  even  numbers  of  which 
2n  is  the  greatest.  Write  three  consecutive  odd  numbers 
of  which  2n-{-l  is  the  greatest. 

115.  It  is  well  to  note  that  if  n  stands  for  any  whole 
number,  then  2n  is  the  symbolic  expression  for  any  even 
number,  since  it  is  exactly  divisible  by  2,  and  2n  +  l  is 
the  symbolic  expression  for  any  odd  number,  since  when 
divided  by  2  the  remainder  is  1. 

EXERCISE  64&. 

Problems 

116.  The  Student  has  already  solved  a  sufficient  num- 
ber of  problems  to  give  him  a  general  idea  of  the  way 
problems  are  solved  in  Algebra.  He  will  find  his  expe- 
rience embodied  in  the  following: 

Directions  for  Solving  Problems  in  Algebra. 
/.  Represent  one  of  the  unknown  numbers,  preferably  the 
one  whose  value  is  asked  for,  by  x. 


PROBLEMS.  133 

//.  Make  symbolic  expressions  to  represent  each  of  the 
other  unknown  numbers  mentioned  in  the  problem. 

III.  Find,  from  the  problem,  two  of  these  symbolic  ex- 
pressions that  are  equal  to  each  other. 

IV.  Solve  the  equation  thus  formed. 

Solve  each  of  the  following  problems  : 

1.  The  sum  of  two  numbers  is  33  and  their  difference 
is  7.     Find  the  numbers. 

Let  «=  one  of  the  numbers, 

then  33—^=  the  other  number. 

And  because  the  difference  of  the  two  numbers  is  7;  therefore, 

a;-(33-;»r)  =  7. 
Removing  parenthesis  x—ZZ-\-x='J. 

Transposing  and  uniting  terms,  2;<r=40, 

whence  a;=20. 

Therefore,  one  number  is  20  and  the  other  13. 

2.  If  56  be  added  to  a  certain  number,  the  result  is 
treble  that  number.     What  is  the  number  ? 

3.  Divide  54  into  two  parts,  such  that  \  of  one  part 
shall  equal  \  of  the  other. 

4.  Divide  100  into  two  parts,  such  that  twice  the 
smaller  part  shall  exceed  the  larger  part  by  8. 

5.  Divide  10  into  two  parts,  such  that  the  sum  of  3 
times  one  part  and  7  times  the  other  part  may  be  42. 

6.  Divide  100  into  two  parts,  such  that  the  first  divided 
by  2  shall  be  equal  to  the  second  divided  by  2. 

7.  Find  a  number  such  that  the  sum  of  its  half  and  its 
third  may  exceed  the  sum  of  its  fourth  and  its  fifth  by  25. 

8.  The  difference  of  two  numbers  is  14,  and  their  sum 
is  52.     Find  the  numbers. 

9.  George  and  Will  have  together  220  marbles.  If 
George  would  give  14  to  Will,  they  would  have  an  equal 
number.     How  many  marbles  has  each  ? 


134  SIMPLE    EQUATIONS. 

10.  I  bought  a  number  of  yards  of  cloth  at  30  cents 
and  an  equal  number  of  yards  at  20  cents,  paying  18.50 
altogether.     How  many  yards  of  each  kind  did  I  buy  ? 

11.  In  a  yard  there  are  chickens  and  rabbits,  and  alto- 
gether they  have  14  heads  and  38  feet.  How  many  rab- 
bits and  how  many  chickens  in  the  yard  ? 

12.  I  bought  3  books  at  a  certain  price,  5  books  at 
double  the  price,  and  4  books  at  treble  the  price  of  the 
first.  In  all  I  paid  $15.  What  was  the  price  of  each 
kind  of  books  ? 

13.  A  man  is  32  years  old  and  his  son  7.  In  how  many 
years  will  the  father  be  just  twice  the  age  of  his  son  ? 

14.  A  clerk  saved  $1280  in  8  years.  The  first  three 
years  he  spent  $360  per  year;  the  next  two  years  he 
spent  $440  per  year,  and  each  year  thereafter  $40  more 
than  the  year  preceding.     What  was  his  annual  salary  ? 

15.  A  man  is  10  years  older  than  his  nephew,  but  15 
years  ago  he  was  twice  as  old  as  his  nephew.  Required 
the  present  age  of  each. 

16.  Two  \yomen  went  to  market  with  equal  amounts  of 
money.  One  spent  $1.20.  The  other  spent  $1.60  and 
returned  home  with  just  ^  as  much  money  as  her  com- 
panion.    How  much  did  each  have  at  first  ? 

17.  Two  farmers,  A  and  B,  owned  a  flock  of  sheep  in 
common,  which  they  agreed  to  divide.  A  took  132  sheep; 
B  took  158  sheep  and  9  lambs,  three  lambs  having  the 
value  of  one  sheep,  and  paid  A  $127.20  in  cash.  What 
was  the  value  of  one  sheep  ? 

18.  A  man  was  engaged  for  80  days  under  the  agree- 
ment that  he  was  to  receive  $4  for  each  day  he  worked, 
but  was  to  forfeit  $1.50  for  each  day  he  was  idle.  He 
received  $276  in  all.     How  many  days  was  he  idle  ? 


PROBLEMS.  135 

ig.  A  servant  was  told  to  buy  a  certain  number  of 
pounds  of  coffee.  If  she  bought  coffee  at  26  cents  a 
pound  she  would  have  18  cents  left,  but  if  she  took  coffee 
at  30  cents  a  pound  she  would  be  short  10  cents.  How 
many  pounds  of  coffee  was  she  told  to  buy,  and  how 
much  money  had  she  ? 

20.  A  number  consists  of  two  digits,  the  sum  of  the 
digits  being  10.  If  the  digits  be  reversed  the  new  num- 
ber is  18  larger  than  the  original  number.  What  is  the 
number  ? 

Let  x=  the  digit  in  tens'  place. 

Because  the  sum  of  digits  is  10;  therefore, 

\Q—x=  the  digit  in  units'  place. 
Hence  the  number  expressed  by  these  digits  is 

10^+10-jr. 
But  the  number  expressed  when  the  digits  are  reversed  is 

10(10- ;r)-|-;r. 
Because  this  is  18  larger  than  the  original  number;  therefore, 

10(10-jc)+x-(10a;+10-;tr)=18. 
Removing  parentheses 

100- lOx+AT- IOjc- 10-t-x=  18. 
Transposing  terms  —10;r-l-x  —  10x+Jc=  18—100+10. 

Uniting  terms  ,         —  18a:=  — 72. 

Therefore,  a;=4,  one  digit. 

Therefore,  10— a  =  6,  the  other  digit. 

Hence  the  number  is  46. 

21.  A  number  consists  of  two  digits,  the  second  of 
which  is  double  the  first.  If  we  reverse  the  digits  the 
new  number  exceeds  the  original  number  by  36.  What 
is  the  original  number  ? 

22.  A  number  consists  of  two  digits  of  which  the  sum 
is  12.  If  we  subtract  18  from  the  number,  we  obtain  the 
number  with  digits  reversed.     What  is  the  number  ? 

23.  A  man  is  32  years  old  and  his  son  2.  In  how 
many  years  will  the  father  be  4  times  as  old  as  his  son  ? 


136  SIMPLE    EQUATIONS. 

24.  A  man  is  32  years  old  and  his  son  8.  In  how  many- 
years  will  the  father  be  4  times  as  old  as  his  son  ? 

25.  A  man  is  32  years  old  and  his  son  11.  In  how 
many  years  will  the  father  be  4  times  as  old  as  his  son  ? 

26.  A  man  is/*  years  old  and  his  son  s.  In  how  many 
years  will  the  father  be  n  times  as  old  as  his  son  ? 

27.  How  long  will  it  take  the  sheriff,  who  can  drive  7 
miles  an  hour,  to  catch  a  thief  who  is  going  5  miles  per 
hour,  and  has  a  start  of  4  hours  ? 

28.  A  train  leaves  a  station  and  travels  at  the  rate  of 
25  miles  per  hour.  Two  hours  later  another  train  follows 
it,  traveling  at  a  rate  of  35  miles  per  hour.  How  long 
before  the  second  train  will  overtake  the  first  ? 

29.  The  body  A  travels  one  yard  a  minute  and  is  pur- 
suing B,  which  is  10  yards  ahead  of  it.  If  A  moves  12 
times  as  fast  as  B,  how  many  minutes  before  A  and  B  are 
together  ? 

30.  At  what  time  between  2  and  3  o'clock  are  the 
hands  of  a  clock  together  ? 

The  minute  hand  moves  over  one  minute  space  in  one  minute.  At 
2  o'clock  the  minute  hand  is  pursuing  the  hour  hand,  which  is  10  spaces 
ahead  of  it.  The  minute  hand  moves  12  times  as  fast  as  the  hour 
hand. 

31.  At  what  time  between  4  and  5  o'clock  are  the 
hands  of  a  clock  together  ? 

32.  A  post  has  i  of  its  length  in  the  earth,  f  in  the 
water  and  13  feet  in  the  air.     How  long  is  the  post  ? 

33.  The  distance  around  a  triangle  is  75  yards.  The 
second  side  is  f  and  the  third  f  of  the  first  side.  What 
is  the  length  of  each  side  ? 

.  34.  A  man  left  half  of  his  estate  to  his  wife,  y^j-f  or 
charity,  f  to  his  children,  and  1400  to  his  servants. 
What  was  the  amount  of  his  estate  ? 


PROBLEMS.  137 

35.  After  a  battle  a  general  ascertained  that  only  f  of 
his  army  was  fit  for  service;  i  of  the  men  were  wounded, 
and  2000  were  dead  or  missing.  What  was  the  original 
strengh  of  his  command  ? 

36.  I  have  three  jars.  The  first  contains  y^^-  and  the  sec- 
ond y\  of  the  amount  contained  by  the  third  jar.  If  I 
fill  the  first  and  pour  the  contents  into  the  second,  it  still 
lacks  1  pint  of  being  full.  What  is  the  capacity  of  each  jar? 

37.  A  mistress  promised  her  servant  1150  a  year  and  a 
new  dress.  The  servant  left  at  the  end  of  10  months, 
and  received  $120  and  the  dress.  What  was  the  value  of 
the  dress  ? 

38.  A  can  do  a  piece  of  work  in  50  days,  B  can  do  the 
same  work  in  60  days,  and  C  can  do  the  same  work  in  75 
days.  How  many  days  will  it  take  them  to  do  the  work 
together  ? 

39.  A  cistern  is  filled  by  three  pipes;  the  first  pipe 
alone  would  fill  it  in  7  hours;  the  second  in  9  hours,  and 
the  third  in  5  hours.  How  long  will  it  take  to  fill  the 
cistern  by  the  three  pipes  together  ? 

40.  The  head  of  a  fish  weighs  2  pounds.  The  tail 
weighs  as  much  as  the  head  and  half  the  body,  and  the 
body  weighs  as  much  as  the  head  and  tail  together. 
What  is  the  total  weight  of  the  fish  ? 

41.  A  can  dig  a  trench  in  ^  the  time  it  takes  B,  and  B 
can  do  it  in  |  the  time  it  takes  C.  By  working  together 
they  do  the  work  in  6  days.  Required  the  time  it  would 
have  taken  each  to  have  done  it  alone. 

42.  There  is  a  group  of  40  persons,  consisting  of  men, 
women  and  children.  The  number  of  women  is  f  the 
number  of  men,  and  the  number  of  children  is  |-  the 
number  of  men  and  women  together.  How  many  each 
of  men,  women  and  children  in  the  group  ? 


138  SIMPLE    EQUATIONS. 

43.  A  person  had  $1000,  part  of  which  he  loaned  at  7 
per  cent,  and  the  rest  at  6  per  cent. ;  the  total  interest 
received  was  163.     How  much  was  loaned  at  7  per  cent.? 

Let  ;f =the  number  of  dollars  loaned  at  7  %. 

Then      1000— :r=-the  number  of  dollars  loaned  at  6  ^. 
Therefore,    ■—— =the  interest  received  at  7  %y 

,    6(1000—^)      ,     .  .      , 

and    —— =the  interest  received  at  6  %. 

Since  the  whole  interest  received  was  $63  ;  therefore, 

7a:      6(1000-^)_ 

lOO"^        100       ~^^' 
whence  a:=300. 

44.  A  man  loaned  a  certain  sum  of  money  at  4J  per 
cent,  for  3  years;  then  he  loaned  the  same  money  at  5|- 
per  cent,  for  2  years;  then  at  5  per  cent,  for  4  years;  then 
at  6  per  cent,  for  3  years.  The  principal  and  interest  for 
the  twelve  years  then  amounted  to  13900.  What  was  the 
original  principal-  ? 

45.  A  principal  of  $4800  having  been  loaned  a  certain 
number  of  years,  amounts  with  simple  interest  to  $6972. 
During  \  of  the  time  it  bore  7  per  cent,  interest;  during 
\  of  the  time  it  bore  7^  per  cent. ,  and  during  the  remain- 
der of  the  time  it  bore  8  per  cent.  For  how  long  a  time 
was  the  money  loaned  ? 

46.  A  man  invested  $1000  at  4  per  cent,  interest,  and  5 
years  afterwards  he  invested  $1800  at  5  per  cent.  In  how 
many  years  will  the  two  sums  be  equal  ? 

47.  A  man  invested  $1100  at  5  per  cent,  interest,  and 
5  years  later  he  invested  $1000  at  7  per  cent  interest. 
How  many  years  from  this  last  date  until  the  prmcipal 
and  interest  of  the  two  investments  will  equal  each  other? 

48.  A  man  invested  a  dollars  at  n  per  cent,  and  d  years 
later  he  invested  b  dollars  at  r  per  cent.  How  long  until 
the  two  sums,  principal  and  interest,  will  be  equal  ? 


CHAPTER  XL 

SIMULTANEOUS  EQUATIONS. 

EXERCISE  65. 

Definitions  and  General  Principles. 

1.  In  the  equation  x-^y=25  what  is  the  valne  of  x  if 
^=4?     If  ^=6?     Ifj^=3?     Ifj|/=i?     If^=f? 

In  the  same  equation  what  is  the  value  of  jk  if  ^=21  ? 
If;t;=19?     If;»;=22?     If  ;r=24|?     If:r=23|^? 

117.  From  this  it  is  plain  that  in  the  equation  x-^y 
=  25,  the  value  of  either  x  ox  y  can  easily  be  found  if  the 
value  of  either  is  give7i,  and  neither  one  of  these  can  be 
found  unless  the  value  of  the  other  is  given. 

It  is  also  plain  that  there  is  an  unlimited  number  of 
pairs  of  values  which  satisfy  this  equation  x-\-y=-1h. 

118.  Equations  in  which  an  unlimited  number  of 
values  can  be  found  for  the  unknown  numbers  are  called 
Indeterminate  Equations. 

2.  In  the  equation  Jir-f  3j/=33  what  is  the  value  of  x 
ifjv=7?     Ifj/=5?     Ifj/=10?     IfjK=4?     Ifj/=t? 

In  the  same  equation  what  is  the  value  of  j^'  if  .r=12  ? 
If:r=18?     IfA'=3?     If  ^=21?     Ifjt:=31. 

Here  as  before  there  is  an  unlimited  number  of  pairs 
of  values  of  x  and  and  y  which  satisfy  the  equation 
.;t+3j/=33,  and  from  this  alone  there  is  no  preference  of 
one  pair  of  values  over  another  pair. 


I40  SIMULTANEOUS   EQUATIONS. 

119.  The  above  illustrations  are  sufficient  to  show  it 
is  trae  in  general  that  the  values  of  two  unknown  mimbers 
cannot  be  determined  from  one  equation  containijig  two  U7t- 
known  numbers  unless  the  value  of  o?ie  of  them  is  given. 

3.  If  Ay—x=l  and  ;r=5,  how  would  you  find  the 
value  of  jK? 

4.  If  2j/-\-Sx=12  and  2x=Q,  how  would  you  find  the 
value  of  x}  of  jj/  ? 

5.  If  4:X—oy=18  and  5>'=10,  how  would  you  find  the 
value  of  _>/  ?  of  ^  ? 

6.  How  much  more  is  x+2y  than  x-\-y}  How  much 
more  is  28  than  25  ?  If  jt:+^=25  and  if  ;»:+2_>/=28,  what 
is  the  value  of  jk  ?  Knowing  the  value  of  y,  how  would 
you  find  x  from  x-\-y=25  ? 

7.  If  2;i;4-:j^=18  and  if  Sx+y=2S,  what  is  the  value 
of  ;»:  ?  Knowing  the  value  of  x,  how  would  you  find  y 
from2;r4-jK=18? 

8.  If  5x—2y=22  and  Sx—2y=10,  what  is  the  value 
of  ;tr  ?  Knowing  the  value  of  x,  how  would  you  find  the 
value  of  y  from  Sx—2y=10  ? 

9.  If  4:X—7y=l  and  4x—5y=S,  what  is  the  value 
of  jK  ?  Knowing  the  value  of  y,  how  would  you  find  x 
from  ix—5y=S  ? 

10.  lix—2y-\-A  and  x=Sy-\-2,  make  an  equation  which 
will  not  contain  the  unknown  numbers.  If  2j/-l-4=3y+2, 
find  the  value  ofy.  Knowing  the  value  ofy,  how  would 
you  find  x  from  x=2y-\-4:  ? 

11.  If  6x=5y-\-ll  and  6x=4y-{-12,  make  an  equation 
which  will  not  contain  the  unknown  number  x.  What 
value  of  y  will  satisfy  the  equation  just  found  ?  Know- 
ing the  value  ofy,  how  can  you  find  x  from  6x=6y-{-ll  ? 


ELIMINATION    BY    SUBSTITUTION.  141 

12.  If  3;^=4jf— 11  and  3>'=j<;— 2,  make  an  equation 
which  will  not  contain  the  unknown  number  jk.  Solve 
the  equation  thus  found.  Knowing  the  value  of  x,  how 
can  you  find  j^  from  Sy=x—2  ? 

120-  In  each  of  the  above  cases  we  have  found  that 
the  values  of  two  uyiknown  numbers  can  be  found  from  two 
equations  C07itaini7ig  the  two  unknown  7imnbers. 

121.  When  two  or  more  equations  containing  several 
unknown  numbers  are  so  related  that  they  are  all  satis- 
fied simultaneously  by  the  same  values  of  the  un- 
known numbers,  the  equations  are  called  Simultaneous 
Equations. 

122.  Of  course,  if  we  can  obtain  from  two  equations 
containing  two  unknown  numbers,  a  single  conditional 
equation  containing  but  one  of  the  unknown  numbers, 
the  value  of  this  unknown  number  can  be  found  in  the 
way  explained  in  the  last  chapter.  When  from  two 
equations  containing  two  unknown  numbers,  we  obtain 
one  equation  containing  but  one  of  the  unknown  num- 
bers, we  are  said  to  Eliminate  the  other  unknown  number. 

We  will  explain  three  methods  of  elimination:  I.  By 
Substitution.  II.  By  Comparison.  III.  By  Addition 
and  Subtraction. 

EXERCISE  66. 

Elimination  by  Substitution. 

123.  Examples  2,  3  and  5  in  the  last  exercise  are  illus- 
trations of  elimination  of  an  unknown  number  by  sub- 
stitution. We  give  a  few  more  examples  worked  by  this 
method. 


142  SIMULTANEOUS    EQUATIONS. 

(1)  Given  2j»:=6y-38  and  _>/  +  23=3jt:  to  find  x  andj^. 

From  the  first  of  these  equations  the  value  of  x  in 
terms  of  y  is  found  to  be 

x=?>j-ld.  (1) 

Substituting   Sy—19  for  x  in  the  second  of  the   given 
equations  we  get 

j/-f23=9j/-57.  ^         (2) 

Transposing  jj/— 9y=— 57  — 23.  (3) 

Uniting  terms  and  dividing  both  sides  by  —1 

8)/=80,  ■  (4) 

whence  jy=10.  (5) 

Substituting  this  value  for^  in  (1)  we  get 

;t:=30-19=ll, 
whence  ;i;=ll  andj/=10. 

To  verify,  we  substitute  these  values  in  the  original 
equations  and  get 

22=60-38  and  10+23=33. 

(2)  Given  7^  +  3^=100  and  3;t:— jf=20  to  find  x  and  jj/. 
From  the  second  equation  the  value  of  y  in  terms  of  x 

is  readily  seen  to  be 

j=3jc-20.  (1) 

Substituting  3;i:— 20  for  jy  in  the  first  of  the  given  equa- 
tions, we  find  7^+9;r-60=100,  (2) 
whence  by  transposing  and  uniting  terms 

16jt:=160.  (3) 

Therefore,  ^=10.        '  (4) 

Substituting  this  value  for  ;r  in  (1)  we  have 

jj/=30-20=10, 
whence  .:r=10  and_>'=10. 


EXAMPLES. 


143 


124.  It  is  easy  to  see  that  the  method  of  elimination 
used  in  these  examples  may  be  applied  in  the  case  of  any 
system  of  simultaneous  equations.  Hence  we  learn  how- 
to  eliminate  an  unknown  number  by  the  Method  of 
Substitution  : 

From  either  equation  express  07ie  of  the  unknown  num- 
bers in  terms  of  the  other,  and  substitute  this  value  in  the 
other  equation. 


EXERCISE  67. 

Examples. 


Solve  the  following  by  the  method  of  substitution  : 


1.  ;^-f-4>/=37. 

2jir+5y=53. 

2.  ;ir+5jj/=573. 

x-\-  jv=181. 

3.  1x-Zy=\m. 
2x-\-  jj/=156. 

4.  Bx+4y=25S. 

y=5x. 

5.  2x-{-9b=llj/. 

;»;_3_>/=0. 

6.  6x—4y=6. 

^x=^ly. 

7.  ;r=3jj/-19. 
;j/=3x-23. 

8.  x+y^UI, 
x-y^lbZ. 


9.  3.r+2j/=:7. 
/Jx—  y=5. 

10.  x-\-5y=47. 
x-h  j/=lo. 

11.  x-\-5y=S5. 
3^+2jK=27. 

12.  ox-{-7y=101. 
7x—  y=^55. 

13.  x=lQ—4y. 
j^=34-4;»:. 

14.  2;t:=ll  +  9>'. 
3;t:-12j/=l5. 

15.  8x—  j/=22. 
2;ir-3^=0. 

16.  Sx-j-4y=18. 
Bx-2y=^70, 


144  SIMULTANEOUS    EQUATIONS. 

EXERCISE  68. 

Elimination  by  Comparison. 

125.  We  give  a  few  examples  of  elimination  of  an 
unknown  number  by  the  method  of  comparison. 

(1)  Given  Sx=7S—y  and  2x=y-{-S2,  to  find  x  and  jy. 

73—0/ 

From  the  first  equation,     x= — ^.  (1) 

From  the  second  equation,    ;r=^       "".  (2) 

Therefore                         73-^^yJ^_  ^^^ 

Clearing  of  fractions,  2(73-_>/)  =  3(j+32).  (4) 

That  is,                          146-2_y=3_r+96.  ,(5) 
Hence,  by  transposing  and  uniting  terms, 

5r=50,  (6) 

whence                                     j/=10.  (7) 

Then,  from  (1),        ^=^ — =21-  (8) 
Hence                       ;r=21  andjK=10. 

(2)  Given9j/+8^=41  and  lljtr— 7jj/=37,  to  find ;»;  andjK. 
From  the  first  equation,    j/= — ^ .  (1) 

From  the  second  equation,   y= ^ .  (2) 

Whence                          — g — = tj •  (3) 

Clearing  of  fractions, 

7(41_8jt:)  =  9(ll;t:-37).  (4) 

That  is,                       287-56;»;=99;i;-333.  (5) 


EXAMPLES. 


145 


Hence,  by  transposing  and  uniting  terms, 

155^=620. 

Therefore  x=i, 

41—32 
Then,  from  (1),       y= — ^ — =1. 


Hence 


x—4:  and  y=l. 


(6) 


126.  These  examples  are  sufficient  to  teach  us  how  to 
eliminate  an  unknown  number  by  the  Method  of  Com- 
parison : 

Express  the  same  unknown  number  in  terms  of  the  other 
from  each  equation  and  equate  the  expressions  thus  found. 


EXERCISE  69. 

Examples. 


Solve  the  following  by  the  method  of  comparison  : 


1.  5:r=63-8>/. 
7;»;=39-3^. 

2.  2^=29— 3:r. 
5>/=20+3;ir. 

3.  3;«r4-^=31. 
5x=15  +  2jy. 

4.  4Ar=19-f7>/. 

5.  16y-3;»:=5. 
5;r4-28>/=19. 

6.  ;r+7=24. 

x—y—lQ. 
10 


7.  5j«r-3j=13.  . 
19x-oy=75. 

8.  7X+  5y=60. 
13j»;-11j/=10. 

9.  3;ir-f2j/=32. 
20;«r-3jj/=l. 

10.  nx-7y=S7. 

8;»;H- 9^=41. 

11.  10.r+  9j=290. 
12;«;-lljK=130. 

12.  x-hy=579. 
;r_v=333. 


146  SIMULTANEOUS   EQUATIONS. 

EXERCISE  70. 

Elimination  by  Addition  and  Subtraction. 

127.  The  elimination   in   the  following  examples  is 
done  by  the  method  of  addition  and  subtraction. 

(1)  Given  x-{-y=67d  and  x— j/=333  find  x  Sindy. 
Adding  the  left  members  and  the  right  members  of 

these  two  equations  together,  we  obtain 

^+jj/=579  (1) 

x-y=SS^  (2) 

2a'=912  (3) 

whence  ;r=456  (4) 

Subtracting  the  members  of  (2)  from  the  corresponding 

members  of  (1),  we  get 

2jj/=246 
whence  y=12o. 

Therefore,  ;r=456  and  j/=123. 

(2)  Given  15:r— 8>'=30  and  Sx+2j'=16  to  find  x  and  j. 
Multiplying  both  members  of  the  second  equation  by 

5,  we  have  for  the  two  equations 

15x-  8y=30  (1) 

15;t:+10r=75  (2) 

By  subtraction                   —18;/=— 45  (3) 

whence                                       J='^i'  (4) 

Hence  from  (1)              15;t:— 20=30.  (5) 

Therfore                                      ^=3  J.  (6) 
Or  the  value  of  x  may  be  obtained  in  another  way. 
Multiplying  both  members  ot    the  second  of  the  given 
equations  by  4,  we  have  for  the  two  equations 

lDx-8y=30  (7) 

12.^4-8;/=  60  (8) 

By  addition                            27ji:=90  *          (9) 
whence                                      ^=|^  or  3|-. 


ELIMINATION.  I47 

(3)  Given  ll;f+12>'=100  and  9;tr+8>'=80  to  find 
X  andj^. 

Multiplying  both  members  of  the  first  equation  by  9, 
and  both  members  of  the  second  equation  by  11,  we 
obtain 

99j^:+108>/=900  (1) 

99.;ir+  88>/=880  (2) 

By  subtraction  20>/=  20  (3) 

whence  jK=l.  (4) 

Substituting   this  value  for  jk  in  either  of  the   original 
equations,  we  find  x=S 

whence  jK=land^=8 

128.  The  above  examples  show  us  how  to  eliminate 
an  unknown  number  by  the  Method  of  Addition  and 
Subtraction : 

Multiply  both  members  of  the  equations  by  such  numbers 
as  will  make  the  7iumerical  coefficieyits  of  one  of  the  u7iknow7i 
munbers  the  same  in  the  resultiiig  equatio7is  ;  the7i  by  addi- 
tion or  stibtradion  we  ca7i  form  a7i  equatio7i  C07itai7ii7ig 
only  the  other  u7ik7iow7i  mwtber. 

Solve  the  following  by  the  method  of  addition  and 
subtraction : 

1.  x+y=ZO.  5.  4;i:+3j/=97. 
;r-j/=6.  7x-{-Sy=127. 

2.  5x-{-7y=17Q.  6.  24x-i-  7y=27. 
5x-Sy=46.  8:r-33j=115.     , 

3.  x-i-5y=^57S.  7.  2;t:+3j=41, 
x+    '=181.  4x+2j/=54. 

4.  x-\-4y=S7.  8.  5x-\-7y=17. 
2x-{-oy=oo.  Tx— 5;'=9. 


148  SIMULTANEOUS   EQUATIONS. 

9.  7:r-3>/=27.  X2.  13j/-  7x=9d. 

5x-6j;=0,  9^-28^=52. 

10.  2Sx-\-15j=4:\,  13.  lGx+  17r=274. 
48jt:+45>/=18.  24;r-105>'=150. 

11.  6;i;-7j/=42.  14.  21x+  8jj/=66. 
7;t:_6>/=75.  28:i:-23>/=13. 

EXERCISE  71. 

Special  Expedients. 

129-  The  student  will  find  that  elimination  by  addi- 
tion and  subtraction  is  in  most  cases  the  shortest  method. 
Occasionally,  however,  an  example  will  be  found  which 
is  more  readily  done  in  some  one  of  the  other  ways. 
Sometimes,  too,  special  expedients  will  still  further  ab- 
breviate the  processes.     We  give  a  few  examples  of  this. 

(1)  Solve  3;r+7^=29,  1x-\-Zy=^Al. 

By   adding   the   members   of  the  two  equations,    we 

obtain                          10^^4-107=70.  (1) 

By   subtracting   the   members  of  second  equation  from 
those  of  the  first,  we  have 

4:x-iy=12.  (2) 

From  (1)  we  get               x-i-y=7,  (3) 

From  (2)  we  get                x—y=S.  (4) 
Whence,  by  addition  and  then  by  subtraction, 
x=o  and_>'=2. 

(2)  Solve -^—g^  =  l,     — -f-^=6. 

To  solve  these  we  must  first  clear  each  of  fractions, 
giving  9j»r— 8jj/=12 

and  14j»r-f  oj/=36 

which  can  now  be  solved  in  any  of  the  usual  ways. 


EXAMPLES.  149 

(3)  Solve =1, 1 =16. 

X     y  X       y 

If  these  be  cleared  of  fractions,  the  resulting  equations 
will  involve  the  product  xy,  and  we  would  have  equa- 
tions of  a  kind  we  have  not  yet  considered.     But  by  con- 
sidering: —  and  -  as  the  unknown  numbers,  we  may  solve 
X         y 

by  the  methods  already  used.  For  example,  by  multi- 
plying both  members  oi  the  first  equation  by  2,  we  get 
for  the  two  equations 

l«-^=2  (1) 

X     y  ^  ^ 

LV^o^ie.  (2) 

X      y  ^  ^ 

28 
By  subtraction  --=14,  (3) 

2 
whence  -=1  or  r=2. 

y 

Therefore  from  (1)  j»:=3. 

We  could  have  eliminated  x  by  dividing  both  members 
of  the  second  given  equation  by  2. 


EXERCISE  72. 

Examples. 

Solve  the  following  by  any  method  : 

1.  x^-y=m.  4-  3;r-f-8>/=19. 
x—y=iS.  ox—  y=l. 

2.  2x-{-y=ll,  5.  3;t:-fcS>/=59. 
Sx-y=4:.  6jr4-5)/=107. 

3.  Sx-h4y=SS.  6.  8x-lDy=S0. 
5x+4y:^107.  2x+  3^=15. 


ISO 


SIMULTANEOUS   EQUATIONS. 


7.  5;i:4-8>'=101. 
9;^;4-2j=95. 

13 

,  2;i: 4- 7^-34=0. 
5;t:-|.9^_51=0. 

8.  Sdx-{-27y=105. 
52;t:4-29j/=133. 

14. 

,   3j/-4;t:-l  =  0. 
18_3.r=4^. 

9.  72^4-14>/=330. 

63:r+   7>/=273. 

15- 

2x+^y=^Al, 
3;r4-2j/=39. 

10.  2x—7y=S. 
4y-9x=ld. 

16. 

19^-21jj/=100. 
21;r-19j/=140. 

II.     8jtr4-3j/=3. 
12:r4-9jj/=3. 

17. 

17jr-i;^=2. 
13jt:  4-17^=236. 

12.  69j/-17-r=122. 
14:ir-23j^/=31. 

18. 

f^4-iy=17. 

19.   10;i:4-|=210. 
10>/-|=290. 

23. 

i4-A=6. 

;r     y 

2-1=0. 

X     J/ 

20.   ~  +  ly^2bl. 
|4-7jt=299. 

24. 

^+^=3. 

X    y 

25. 

1+2=10. 

X    y 

i+^=20. 

26. 

^+^=19. 

^-^=13. 
X    y 

EXAMPLES.  151 


27.  4+^=—.  31.  — +f:=7. 


4  •  -^  •  3.r  '  5j/ 

tv  J 1^ 

5"*  ^x     10/ 


28.  o+^=o-  32.  -^-+-^=5. 


8  ■  2     3*  -^         8      ■     6 

2"^3     6 


•^-fr^=I  :^i^_£:i^=io. 


29.  ?+|=31.  33.  ?:^+3=^. 


4  '  6~  ^* 

X    y 

3  ""8'^  2*  ^        4     ~2'4' 


X    y_\  3     .^-2^/^^  ^  J 


1.      1_11  4.r+5j 

;»;"^j~30-  ^'*-       40 

111  1     2jf-j/ 


X     y    30'  .  2         3         ^-^' 

35.-4 4-  =  ^-V- 

2.r+j^     9jr— 7_3jK+9     4;>:4-5y 
2  8     ~     4  16     • 

36.  -y ^=3^-5. 


152  SIMULTANEOUS    EQUATIONS. 

EXERCISE  73. 

Simultaneous  Equations  Containing  Three  Unknown  Numbers. 

130.  If  we  have  two  equations  containing  three  un- 
known numbers,  such  as 

we  can  eliminate  one  of  these  unknown  numbers  by  the 
methods  already  explained,  giving  one  equation  contain- 
ing two  unknown  numbers.  Thus,  in  this  particular 
case,  by  multiplying  the  members  of  the  second  equation 
by  2  and  subtracting,  we  would  find 

7r-192'=-17. 
Since  an  indefinite  number  of  values  will  satisfy  one 
equation   containing  two  unknown  numbers;  therefore, 
an  indefinite  number  of  sets  of  values  will  satisfy  two 
equations  containing  three  unknown  numbers. 

Suppose,  however,  that  we  have  three  equations  con- 
taining three  unknown  numbers,  as 

2,r+3y-5^=9  (1) 

jr-2j^+7<?=13  (2) 

Zx-  y-1z=^.  (3) 

Eliminating  x  from  (1)  and  (2)  as  explained  above,  we 
get  7>/-19^=~17.  (4) 

Multiplying  both  members  oi  (2)  by  3,  and  subtracting 
from  (3),  we  obtain 

5r-23j=-31.  (5) 

Now  we  can  eliminate  _y  from  (4)  and  (5)  by  multiplying 

both  members  of  (4)  by  5,  and  both  members  of  (5)  by  7, 

giving  35y—  95^=— 85  (6) 

35^- 161^= -217,  (7) 


THREE  UNKNOWN  NUMBERS.         1 53 

whence,  by  subtraction,  66^=132,  (8) 

whence  z=2. 

Substituting  2  for  z  in  (4) 

7y-38=-17,  (9) 

whence  j^'=3, 

and  substituting  j=3  and  ^=2  in  (1),  we  find  x—b. 
Therefore,  ;r=5,  jj/=3,  and  z=2. 

Here  we  notice  that  we  have  been  able  to  find  the 
values  of  three  unknown  numbers  from  three  equations. 

131.  It  is  evident  that  we  can  proceed  in  a  similar 
way  in  any  case  of  three  equations  containing  three  un- 
known numbers.  That  is,  to  solve  three  simultaneous 
equations  containing  three  unknown  numbers  : 

/.  Obtain  froyn  two  of  the  equations  a7i  equation  which 
contaifis  only  two  of  the  tinknown  numbers^  by  any  method 
of  elimination. 

II.  Fro?n  the  third  given  equation  and  either  of  the 
former  two,  obtaiji  anothe>  equatiofi  which  contains  the  same 
two  U7iknown  numbers. 

III.  Fro7n  the  two  eqtiations  coniaiyiing  two  unknown 
numbers  thus  fou7id,  fina  the  values  of  these  unknown 
numbers. 

IV.  By  sicbstituting  these  values  in  one  of  the  give7i 
€quatio7is,  the  value  of  the  re7nai7iing  unk7iown  number 
may  befo:  nd. 

132.  We  will  further  illustrate  this  subject  by  work- 
ing a  few  examples.  It  should  be  observed  that  while  it 
makes  no  difference  which  one  of  the  unknown  numbers 
we  eliminate  first,  yet  the  work  is  often  lessened  by  the 
selection  for  this  purpose  of  that  one  of  the  unknown 
numbers  whose  numerical  coefficients  have  the  smallest 
L.  C.  M. 


154  SIMULTANEOUS   EQUATIONS. 

(1)  Solve  4x-  5y-\-  ^=6.  (1) 

7;i:-lljj/+2^=9.  (2) 

x+    j^/+32'=12.  (3) 

The  unknown  number  z  has  the  smallest  numerical 

coefficients,  and  it  is  easier  to  eliminate  it  than  any  of 

the  other  unknown  numbers.     Multiplying  both  members 

of  (1)  by  2,  we  have 

8^-10)/+ 2^=  12.  ^  (4) 

Subtracting  (2)  from  this,     x-j-y^S.  '  (5) 

Now  multiply  both  members  of  (1)  by  3,  giving 

12x-15>/+3^=18.  (6) 

Subtracting  (3)  from  this,  we  find 

ll.r-16>/=6.  (7) 

We  have  now  to  find  the  values  of  x  and  j^  from  (5) 
and  (7).     Multiply  both  members  of  (5)  by  11,  giving 
ll^+llj/=33.  (8) 

Subtracting  (7)  from  this,  we  find 

27y=27,  (9) 

whence  j=l. 

From  (5)  x=2 

and  from  (3)  2  +  1  +  3^=12,  whence  ^=3. 
Therefore  x=2,    j=l,  and  ^=3. 

The  student  may  verify  these  in  the  original  equations. 

(2)  Solve  A'-fj=5.  (1) 

y-hz=7.  •  (2) 

x+z=6.  (3) 

This  is  quickly   solved    by  special  expedient.     Thus 

add  the  members  of  the  three  equations  together,  giving 

■      2x-\-2y+2z==lS 
or  x-\-  y-\-  ^•=9.  (4) 

From  (1)  x-\-y=b,  therefore  from  (4)  ^=4. 
From  (2)  y-\-2=l^  therefore  from  (4)  x=2. 
From  (3)  x-^2=Q>,  therefore  from  (4)  y=Z. 


EXAMPLES.  155 

(3)  Solve -+-+-=4.  (1) 

X     y     z 

X    y     z 

^+L2_10=4.  ■  ■  (3) 

X     y      z 

Here  we  should  consider  -,     — ,  and-  as  the  unknown 

X      y  z 

numbers.     Subtracting  (1)  from  twice  (2),  we  get 

^^^=4.  (4) 

y      z 

Subtracting  (3)  from  three  times  (2),  we  get 

y      z 

We  are  now  to  find  -  and  -  from   (4)  and  (5).     Sub- 
y  z 

tracting  (4)  from  (5), 


?-. 

whence 

^=5. 

From  (4) 

^=4. 

From  (1) 

x^Z. 

EXERCISE  74. 

Examples, 

Solve  the  following  simultaneous  equations  : 

1.  x-\-y=S7.  3.  x-h  y+  z=ZO, 
x+z=2o.  Sx-^4y+2z=50. 
y-\-  ^=22.  27;»;+9>/+3^=64. 

2.  2x+y  =5.  4.  5j*r4-7y—  2^=13. 

;»:4-3^=16.  8x-{-Sy-\-     ^=17. 

6y-  z=10.  x-4y+10z=2S. 


156  SIMULTANEOUS   EQUATIONS. 


5.  5;^-3y=l. 

7.     3;i:+2>/+  ^=23. 

9y-2z=12. 

5;»:4-2jK+4^=46, 

8;^+3^=17. 

10;i:+5>/H-4-3'=75. 

6.  x+y-z=17. 

8.  4x—2y+52=lS. 

x-\-2-y=lZ. 

2x+4:y-Bz'-==22. 

yJ^2-X=1. 

6x+7y-  ^=63. 

9.     X 

+    ^  +  ^ 

1^=23. 

Zx 

_  y^2z=\\. 

X 

•+4y~ 

^=4. 

0  M=^^- 

X4.  i+i=l. 

X    y 

H='- 

i+i=2. 

12+7=^- 

V     z     2* 

'•.-fe- 

2,13 

^5-  ^+ri- 

^r- 

i-^=2. 

^r^- 

i+i=i 

..  f +1+1=62. 

16.  — \ =1. 

X     y     z 

f+f+l=^^- 

^+i+^=24. 
X    y     z 

f+M=^«- 

X    y     z 

3.  258-H+f- 

3_4      1_38 
X     hy     z      5  * 

304-£=f+|. 

Sx     2y^z      6* 

296-f=£+|. 

4       1      4     161 
5x     2y  '  ^"~  10  • 

LITERAL    SIMULTANEOUS    EQUATIONS. 


157 


EXERCISE  75. 

Literal  Simultaneous  Equations. 

Solve  the  following  : 

I.  x-\-y=2a  and  x—y=2d. 


By  addition, 

2x- 

:2^+2/^ 

whence 

X- 

:    ^Z+    <J. 

By  subtraction, 

2y- 

:2^-2^ 

whence 

J= 

:    a-    b. 

2.  x-hy-=Sa-2d. 

, 

4.  «(3«  +  ;r)  =  ^(^H-j/). 

x—y=2a  —  U. 

ax-\-2by=^d. 

3.  2x—Sy-=dd—a. 

5.   rt.r+4y=^. 

Sx-2y=a  +  d5. 

w.r+r>/=x. 

6.   adx+cdy=2. 

d-d 
ax-cy^-^^^ 

8.  f-^=^ 

1   ,  1 

X     y 

71       \ 

9.  -4— =«. 

1      1 
— r. 

X     y 

X     y 

10.   :rH-jj/=2«. 

13.  «.r4-<^j^— <r2'=2«^. 

x-Vz^2b. 

by-\-cz—ax=^2  be. 

y-\-z=2c. 

cz + «jt"—  by^=  2ac. 

II.  ajr+_>'=r. 

14.  .r— «;r+j=0. 

^-^=^. 

^y— /J  +  -3'=0. 

bx^-z=t. 

x-^z=t. 

12.  ;«;+jj/H-2'=^. 

15.  jr+«^+«2^+«^  =  0. 

;i;— ^+2'=^^. 

jr+^^+/^-^+^^=0. 

;c+J^— 2'=r. 

^+0'+<:^'S' 4-^^=0. 

X 

n 

y~ 

r 

y^ 

J- 

z 

9 

1 

1 

17.  -= 

-a- 

— 

'      X 

y 

1 

1 

— = 

^b- 

y 

z' 

1 

1 

— = 

-c— 

z 

X 

158  SIMULTANEOUS    EQUATIONS. 


a      b      c 

16.  x-\-y-\-z=a.  18.  — I =n. 

•^  X    y     z 

a      b     c  _ 

X    y     z 

X    y     z 

19-  -+-+-=3. 
X    y     z 

X    y     z 
s2a     b     c ^ 

X     y     z 


EXERCISE  76. 

Problems  Producing  Simultaneous  Equations. 

1.  The  sum  of  two  numbers  is  70,  and  their  difference 
is  24.     Find  the  numbers. 

Problems  exactly  similar  to  this  have  been  worked  with  the  use  of 
one  unknown  number.  We  will  now  work  it  using  two  unknown 
numbers. 

Let  ic=the  first  number, 

and  let  jj/  =  the  second  number. 

Then,  because  the  sum  of  the  numbers  is  70,  and  the  difference  of 
the  numbers  is  24  ;  therefore, 

■^+7=  70, 
and  re— 7=24. 

Solving  these,  we  find  ic=:47  and  7=23. 

2.  Find  a  fraction  such  that  if  we  add  1  to  the  numer- 
ator, it  becomes  equal  to  \,  but  if  we  add  2  to  the  de- 
nominator, it  becomes  equal  to  \. 

Let  x=the  numerator  of  the  required  fraction, 

and.let  jj'=the  denominator  of  the  required  fraction. 

Because  the   fraction  with  1  added    to  the    numerator  equals  \, 

therefore,  -—-=—.  (1) 

y         1  ^  ' 


PROBLEMS.  159 

Because  the  fraction  with  2  added  to  the  denominator  equals  ^, 
X        1 

therefore,                                        —r^=-7:-  (2) 

From  (1)  and  (2)  we  get,  by  clearing  of  fractions, 

2a:+2=j  (3) 

^x=y^2.  (4) 

Eliminating/  we  get  x=4,  whence  from  (3)^  =  10.  The  fraction  is, 


3.  Says  A  to  B:  "  Give  me  $49  and  I  will  have  just  as 
much  money  as  you."  Says  B  to  A:  "  Give  me  $49  and 
I  will  have  3  times  as  much  money  as  you. ' '  How  much 
has  each  ? 

Let  :c=the  number  of  dollars  A  has. 

Then  j=the  number  of  dollars  B  has. 

If  B  gives  A  $49,  A  would  have  x-|-l9  dollars  and  B  would  have 
;'  — 49  dollars.  Because  they  would  then  have  equal  amounts,  there- 
fore, x4-49=j-49.  (1) 

If  A  gives  B  $49,  B  would  have  y-\-^2  dollars  and  A  would  have 
x-49  dollars.  Because  B  would  then  have  3  times  as  much  as  A, 
therefore,  3(jt -49)  =  r+49.  (2) 

Eliminating  J  from  (1)  and  (2),  we  get 

2j:-196  =  98,  (3) 

whence  x=\Al.  (4) 

From  (1)  we  then  find  ^=245. 

Therefore,  A  has  $147  and  B  has  $245. 

4.  A  man  bought  two  kinds  of  cloth,  7  yards  of  the 
first  kind  and  11  yards  of  the  second  kind,  and  paid  $47 
for  both.  If  he  had  bought  11  yards  of  the  first  kind  and 
7  yards  of  the  second  kind  he  would  have  saved  $4. 
What  was  the  price  per  yard  of  each  kind  of  cloth  ? 

5.  Find  a  fraction  such  that  when  1  is  added  to  both 
numerator  and  denominator,  it  equals  \,  but  when  3  is 
subtracted  from  numerator  and  denominator  it  equals  \. 

6.  Find  a  fraction  such  that  when  11  is  taken  from 
both  numerator  and  denominator  it  equals  f,  but  when  12 
is  taken  from  both  numerator  and  denominator,  it  equals  \. 


l6o  SIMULTANEOUS    EQUATIONS. 

7.  A  fraction  is  such  that  if  I  add  1  to  the  numerator, 
the  fraction  equals  unity ;  but  if  I  double  the  first  frac- 
tion and  add  1  to  the  denominator,  the  value  of  the 
fraction  equals  unity.     What  is  this  fraction  ? 

8.  A  number  is  formed  of  two  digits  of  which  the  dif- 
ference is  3.  If  the  digits  be  reversed,  a  number  is 
obtained  which  is  7-  of  the  original  number  ?  What  is  the 
original  number  ? 

Let  X  represent  the  digit  in  tens'  place,  and  y  represent  the  digit  in 
units'  place. 

9.  A  farmer  sold  to  one  person  9  horses  and  7  cows  for 
$1500,  and  to  another,  at  the  same  price,  6  horses  and  13 
cows  for  the  same  money.     What  was  the  price  of  each? 

10.  The  sum  of  two  digits  which  form  a  number  is  9. 
If  we  subtract  3  from  each  digit,  the  result  is  6  more 
than  half  the  original  number.  What  is  the  original 
number  ? 

11.  Two  masons,  A  and  B,  are  building  a  wall,  which 
they  could  finish,  working  together,  in  12  days.  A  works 
3  days  and  B  2  days,  when  the  wall  is  \  done.  How 
long  would  it  have  taken  each  to  have  built  the  wall  ? 

Let  a;=number  of  days  it  would  take  A, 

and  y=number  of  days  it  would  take  B. 

In  one  day  A  builds  —  of  the  wall,  and  B  — .  But  together  they  build 

^  of  the  wall  in  one  day.     Therefore, 

3  2 

In  3  days  A  builds  —  of  the  wall,  and  in  2  days  B  builds—  of  the 

X  y 

wall.     Therefore,  from  the  problem, 

l+f  i  <^) 

Solving  the  simultaneous   equations  (1)  and  (2),  we   can    find    the 
values  of  x  and^. 


PROBLEMS.  l6l 

12.  A  and  B  can  together  do  a  certain  work  in  30 
days  ;  at  the  end  of  18  days,  however,  B  is  called  off  and 
A  finishes  it  alone  in  20  days  more.  Find  the  time  in 
which  each  could  do  the  work  alone. 

13.  A  cistern  holding  4500  gallons  is  filled  by  two 
pipes.  If  the  first  pipe  be  opened  3  minutes  and  the 
second  pipe  1  minute,  400  gallons  will  run  into  the  cis- 
tern ;  but  if  the  first  pipe  be  opened  1  minute  and  the 
second  7  minutes,  600  gallons  will  run  in.  How  much 
water  does  each  pipe  carry  in  one  minute  ?  How  long 
will  it  take  both  pipes  to  fill  the  cistern  if  they  are  opened 
together  ? 

14.  A  cistern  can  b-^  filled  by  two  pipes.  If  both  pipes 
be  opened  for  15  minutes  they  will  fill  ^  of  the  cistern  ; 
but  if  the  first  pipe  be  opened  for  12  minutes  and  the 
second  for  20  minutes,  A  of  the  cistern  will  be  filled. 
How  long  will  it  take  each  ot  the  pipes  to  fill  the  cistern 
when  opened  alone  ? 

15.  A  man  receives  $2160  yearly  interest  on  his  cap- 
ital. If  he  had  loaned  the  same  capital  at  ^  per  cent, 
higher  interest  he  would  receive  $240  more  interest  each 
year.  Find  the  amount  of  his  capital  and  the  rate  per 
cent. 

16.  A  man  has  two  sums  of  money  at  interest,  the 
first  at  4  and  the  second  at  5  per  cent.  Together  they 
bring  in  $3000  interest  annually.  What  is  the  amount 
of  money  loaned  at  each  rate  ? 

17.  A  man  has  two  sums,  one  of  $10000  and  another 
of  $15000,  at  interest,  and  receives  therefrom  $1200  an- 
nually. If  the  first  sum  had  been  loaned  at  the  rate  that 
the  second  bore  and  if  the  second  sum  had  been  loaned 
at  the  rate  that  the  first  bore,  he  would  have  received 
$25  less  per  year.     At  what  rates  were  the  sums  loaned  ? 


l62  SIMULTANEOUS    EQUATIONS. 

i8.  A  and  B  can  do  a  piece  of  work  in  12  days  ;  B  and 
C  can  do  it  in  20  days  ;  A  and  C  in  15  days.  How  long 
will  it  take  each  to  do  the  work  alone  ? 

ig.  A  cistern  is  filled  with  three  pipes.  The  first  and 
second  will  fill  it  in  72  minutes,  the  second  and  third  in 
120  minutes,  and  the  first  and  third  in  90  minutes.  How 
long  would  it  take  each  one  of  the  pipes  to  fill  it  ? 

20.  Three  cities,  A,  B,  and  C,  are  at  the  corners  of  a 
triangle.  From  A  through  B  to  C  is  82  miles  ;  from  B 
through  C  to  A  is  97  miles  ;  from  C  through  A  to  B  is  89 
miles.  How  far  are  the  cities  A,  B,  and  C  from  one 
another  ? 

21.  A  certain  number  consists  of  three  digits,  whose 
sum  is  15.  If  the  first  two  digits  be  reversed  the  num- 
ber becomes  180  larger,  but  if  the  last  two  digits  be  re- 
versed, the  number  becomes  but  18  larger.  What  is  the 
number  ? 

22.  The  sum  of  three  numbers  is  70.  The  second 
divided  by  the  first  gives  2  for  the  quotient  and  1  for  the 
remainder,  but  the  third  divided  by  the  second  gives  3 
for  the  quotient  and  3  for  the  remainder.  What  are  the 
numbers  ? 

23.  There  was  a  family  of  boys  and  girls  and  when 
they  were  asked  how  many  there  were  in  the  family  one 
of  the  boys  said:  "I  have  just  as  many  brothers  as 
sisters."  But  one  of  the  girls  said:  "  I  have  twice  as 
many  brothers  as  sisters. ' '  How  many  boys  and  girls  in 
the  family? 

24.  It  takes  72  English  and  51  German  yards  together 
to  make  100  meters.  Also  48  English  and  84  German 
yards  make  100  meters.  How  many  inches  (English)  in 
the  meter  ?  How  many  inches  (English)  in  the  German 
yard  ? 


CHAPTER  XII. 

POW'ERS  AND  ROOTS 

EXERCISE  77. 

Powers  of  Monomials. 

133.  The  process  of  raising  a  number  or  expression 
to  any  power  is  called  Involution.  We  have  already 
learned  that  «"«''= ^"+'',  (1) 

where  ii  and  r  are  any  positive  whole  numbers,  but  where 
a  may  be  either  a  whole  number  or  a  fraction,  and  either 
positive  or  negative. 

Multiply  both  numbers  of  equation  (1)  by  a'  and  we 
obtain  a"  a'' a' ==  a*'^'' a\  (2) 

But  the  right  side  of  equation  (2)  is  the  product  of  two 
powers  of  the  same  letter,  and  hence  from  what  we  have 
learned  before,  the  right  side  of  (2)  equals  a"+''+^. 

Hence,  «"a''«^=a''+''+\  (3) 

1.  In  the  same  w^ay  show  that 

2.  Can  you  find  in  the  same  way  the  product  of  more 
than  four  factors,  each  one  of  which  is  a  power  of  <2  ? 

3.  Can  you  find  the  product  of  any  number  of  factors, 
each  one  of  which  is  a  power  of  the  same  number  ? 

4.  State  then  what  the  product  of  aiiy  number  of 
powers  of  the  same  number  is  equal  to. 

5.  In  formula  (1)  can  the  exponents  n  and  rbe  equal 
to  each  other  or  must  they  be  different  ? 

If  you  cannot  readily  answer  this  question,  turn  back  to  page  55 
and  sec  how  this  formula  was  proved,  and  then  the  answer  will  be  clear 


164  POWERS^  AND    ROOTS. 

6.  What  does  a'^a^  equal?  What  then  does  {a'^Y  equal? 

7.  What  does  a^a^  equal  ?  What  then  does  (a^)^  equal  ? 

8.  What  does  a'^a'^  equal?  What  then  does  («*)^  equal  ? 

9.  What  does^:''^^  equal?  Whattlien  does  (a ^)''^  equal  ? 

10.  What  does  a"a"  equal  ?  What  then  does  (a")^  equal  ? 

11.  In   formula   (3)  can  the  exponents  n,  r  and  ^  be 
equal  to  one  another  ? 

12.  What  does  a'^a'^a^  equal?     What  then  does  (<^^)^ 
equal  ? 

13.  What  does  a^a'^a^  equal?     What  then  does  (a^)^ 
equal? 

14.  What  does  a^a^a^  equal?     What  then  does  {a^Y 
equal  ? 

15.  What  does  a^a^a^  equal?     What  then  does  (a^Y 
equal  ? 

16.  What  does  a"a''a"  equal?     What    then  does  (a"Y 
equal  ? 

17.  What  does  «'V^"a"  equal  ?     What  then  does  (^")* 
equal  ? 

18.  What  does  a"a"a"a"a"  equal  ?    What  then  does  (a")  ^ 
equal  ? 

19.  What  does  the  product  of  r  factors  each  of  which  is 
a"  equal  ? 

20.  What  then  does  {a"y  equal  ? 

134.  These  questions  lead  to  the  general  formula, 

[a"  Y  -W''. 
The   truth   expressed    by   this  formula   n:a}-   be   ex- 
pressed in  words  thus:      The  rth  power  of  the  n  th  power 
of  a  number  is  equal  to  the  nr  th  power  of  that  yiiiviber. 


POWERS    OF    MONOMIALS.  165 

135.  Let  us  now  seek  a  power  of  the  product  of  dif- 
ferent numbers. 

Consider,  first,  the  square  of  the  product  of  several 
numbers. 

(abY =abab=aabb=a'^  b"^ ', 

also  (abcY  ^abcabc^aabbcc^a"^  b'^  c*^ , 

21.  In  the  same  way  show  that 

also  that  (abcde')'^  =  a-b'^c'^d'^e'^. 

22.  Would  a  similar  result  hold  if  there  were  more  than 
five  factors  in  the  product  to  be  squared  ? 

23.  Justify  the  following  statement : 

T/ie  square  of  the  prodiid  of  any  number  of  numbers  is 
equal  to  the  product  of  the  squares  of  those  yiumbers. 

Consider,   next,   the  cube  of  the   product  of  several 

factors. 

(^abY=ababab=aaabbb=a^b^  ; 
also  (abc)  ^  =  abcabcabc=-aaabbbccc=  a'^b'^c^. 

2/[.  In  the  same  way  show  that 

(abedy=aH^c^d^; 
also  that  (abcde)^=a^b^c^d^e^. 

25.  Would  a  similar  result  hold  if  there  were  more 
than  five  factors  in  the  product  to  be  raised  to  the  third 
power  ? 

26.  Justify  the  following  statement: 

The  cube  of  the  product  of  ajiy  number  of  numbers  is 
equal  to  the  product  of  the  cubes  of  those  numbers. 

27.  The  fourth  power  of  the  product  of  any  number  of 
numbers  is  equal  to  what  ? 


1 66  POWERS   AND    ROOTS. 

28.  In  the  n  th  power  of  the  product  of  any  number  of 
numbers  the  first  number  appears  how  many  times  as 
a  factor  ?  The  second  number  appears  how  many  times 
as  a  factor  ?  Any  one  of  the  numbers  appears  how  many 
times  as  a  factor  ?  Therefore,  the  n  th  power  of  the 
product  of  any  number  of  numbers  equals  what  ? 


G) 


136.   IvCt  us  now  seek  a  power  of  the  quotient  of  two 
numbers. 

a  a     aa      d^ 

,  (a\  ^      a  a  a     aaa     a^ 

^''°  Kb)  =-b  '  -Tfib^b^- 

29.  In  the  same  way  show  that 

,4 


©' 


a 

A4"' 


also  that  (-3'=|t: 


also  that 


\b)    ~~¥' 


137.  Thus  we  may  raise  to  any  desired  power, 
First,  a  power  of  a  number  ; 

Second,  the  product  of  several  numbers  ; 

Third,  the  quotient  of  two  numbers. 

These  three  cases  include  every  monomial  that  can  be 
proposed.  Hence  any  monomial  can  be  raised  to  any 
required  power  by  the  methods  already  given. 

138.  As  any  even  power  of  a  monomial  having  a  — 
sign  is  the  product  ot  an  even  number  of  subtractive 
terms,  it  follows  that  any  even  power  of  a  monomial 
having  a  —  sign  is  a  monomial  having  a  -f-  sign  or  no 
sign.  Thus,  (-2)2= +4,  (-3)-^=  SI,  {^-lab'^y 
=  G4«6^i^     etc. 


EXAMPLES. 


167 


139.  Again,  as  any  ^^<a?  power  of  a  monomial  having  a 
—  sign  is  the  product  of  an  odd  number  of  subtractive 
terms,  it  follows  that  any  odd  power  of  a  monomial 
having  a  —  sign  is  a  monomial  having  a  —  sign.  Thus, 
(-2)3  =  -8,     (-3)^  =  -243,     (-2«^2)7  =  _128^7^i4_ 


Write   down  the  value  of  each  of  the  following  ex- 
pressions: 


I.  (Sad'^y. 

2.  i^xyy. 


3.  (2aH^-)K 

4.  (-3^:^2)2^ 

5.  (-2^3^2)2. 

6.  (5aby, 

7.  (3aU'2).2 


10 


II 


12 


13 


/5xz\  2 
\7j)   ' 
/2x-zY 

/4x^Y 
\2uv)    ' 

\  'Sax  )   ' 


.» (pi)', 

20. . 


dax 
labc''  \  '"  (lax  \  *  /     1     \ '      /     1     \ ' 


30.    l-t.\^m\ 
^       \xyz)        \abc  J 


1 68  POWERS    AND    ROOTS. 

EXERCISE  78. 

Square  of  a  Binomial. 

140.  By  actually  multiplying  a  +  b  hy  a-^b  we  find 
that  (a  +  dy^a''-^2ad-j-d\ 

As  a  and  d  may  stand  for  any  numbers  whatever,  we 
may  say  that 

The  square  of  the  sum  of  any  two  numbers  is  equal  to 
the  square  of  the  first  nujnbery  plus  twice  the  product  of  the 
first  aiid  second,  plus  the  square  of  the  second. 

141.  Again  by  actually  multiplying  a—b  by  a—b  we 
find  that  {a-by=^a''-'-1ab-\-b''. 

As  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  say  that 

The  square  of  the  difference  of  a7iy  tzvo  numbers  is  equal 
to  the  square  of  the  first  7iu7nber,  minus  twice  the  product  of 
the  first  and  second,  plus  the  square  of  the  secorid. 

The  two  statements  in  italics  are  so  important,  and  are  used  so 
often,  that  they  should  be  thoroughly  familiar  to  the  student. 

According  to  the  two  statements  in  italics,  write  down 
the  square  of  each  of  the  following  binomials  : 


I. 

x-\-y. 

7. 

Sx-2y\ 

13.  -+^. 
"^   y     X 

2. 

Ix-^Z. 

8. 

4ab-\-2a'^. 

1  ,  1 
^4-  2  +  3- 

3. 

x^--\-ab. 

9- 

5.-|. 

3  ,  1 

15.  2  +  3-^-^' 

4. 

uv-\-w. 

10/ 

^- 

16.  204-5. 

5- 

rx^-t"-. 

II. 

u 

w . 

V 

17.  100-1. 

6. 

2xy^^\ 

12. 

4xy—^. 

18.  30  +  7. 

CUBE    OF    A    BINOMIAL.  169 

19.40-3.  23.   ax^+(^^^y,  27.  ix^-^,, 

20.  x*-Sy\  24.  (Sad^-y  +  2aH.  28.  (2^^)24./ IV 

21.  10;t:+j.  25.   (5^0'- (^)'  29.  (3^2)3.^(5^)2^ 
X      a             -    2;n'5'      2^5/  02.'    9   ^ 

EXERCISE  79. 

Cube  of  a  Binomial. 

142.  In  the  last  exercise  we  found  that 

{a-^by  =  a''^-1ab-^b''. 
Multiply  each  member  of  this  equation  by  a-^b,  and 
we  obtain 

{a-^bY^a^^-Za''b^-^ab''^-b^. 

The  student  should  actually  go  through  the  work  of  multiplying 
the  second  member  of  the  first  equation  by  a-\^b  to  see  that  this  re- 
sult is  correct. 

As  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  say  that 

The  cube  of  the  su7n  of  any  two  numbers  is  equal  to  the 
cube  of  the  firsts  plus  3  thnes  the  squai'e  of,  the  first  multi- 
plied by  the  second,  plus  3  times  the  first  Tnultiplied  by  the 
square  of  the  seco7id,  plus  the  cube  of  the  second. 

143.  In  the  last  exercise  we  found  that 

(a^by  =  a''-2ab+b\ 
Multiply  each  member  of  this  equation  by  a—b,  and 

we  obtain 

{a-by=a^~-ZaH  +  ^ab''-b^. 

As  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  say  that 


I/O  POWERS    AND    ROOTS. 

The  cube  of  the  difference  oj  two  numbers  is  equal  to  the 
cube  of  the  first,  fninus  3  times  the  sguare  of  the  first  mul- 
tiplied by  the  second,  plus  3  times  the  first  multiplied  by  the 
square  of  the  second,  mirius  the  cube  of  the  second. 

The  two  statements  in  italics  are  so  important  and  so  frequently 
used,  that  they  should  be  thoroughly  familiar  to  the  student. 

By  means  of  the  two  statements  in  italics  write  down 
the  cube  of  each  of  the  following  binomials  : 


I. 

x^y. 

8. 

3a^2+2jt;. 

15- 

40-3. 

2. 

x—ly. 

9. 

2y''^-V?>z\ 

16. 

x^-^a^-y. 

3. 

ab-\-c. 

10. 

4:x—3ab. 

17- 

nt'^—r'^s'^. 

4. 

c—ab. 

II. 

5ab—Sac. 

18. 

(a^2)2+(2x)2. 

5- 

x-y. 

12. 

10  +  5. 

19. 

(2xy-\-2x\ 

6. 

2ax-^b. 

13- 

100-1. 

20. 

a-{-a. 

7. 

2ax-2b. 
3.+f. 

14. 

30  +  7. 

21. 
34. 

i-ixy^'^y. 

22. 

28. 

uv-^ 0. 

uv 

x^y^ 
a      a 

23. 

y      b 

29. 

lo-fo- 

35- 

"■+(*)'• 

24. 

^x"^     2yy 
'^y'^'Zx' 

30. 

^n      Zs 
Zs      4n ' 

36. 

Ax" 

w. 

31. 

n'^x'^     71-x'^ 

37. 

-.1.. 

25- 

2x         2n   ' 

26. 

1       1 

ab     ac 

32. 

abc 
rst 

38. 

a 

27. 

a      b 
b  ■  a 

33. 

?>x     2yz^ 

y       X 

39- 

«2     22 
¥      a' 

SQUARE    OF    A   POLYNOMIAL.  I/I 

EXERCISE  80. 

'         Square  of  a  Polynomial. 

144.  It  is  so  often  necessary  to  square  a  polynomial 
that  it  is  well  to  have  a  way  of  writing  down  the  result 
without  being  obliged  every  time  to  go  through  all  the 
work  of  multiplying. 

145.  Let  us  first  take  a  polynomial  of  /our  terms  and 
find  its  square  by  actual  multiplication. 

a-\-b-\-c-Vd 

«2-f   ab-{-   ac-{-   ad 

ab  +<^2_j_   jjcJ^   Id 

ac  +   be  -\-c^-\-  cd 

ad  4-   bd        +   cd^-d"^ 

a"^  -\-2ab-\-2ac+2ad+  b''  ^2bc+2bd+c^  +  '2cd+d'' 
Slightly  changing  the  order  of  the  terms  in  this  result, 

we  may  write 

{a  +  b+c-^dy 
=  a''  -\-b''  +C''  ^d"-  +  2ab+'lac+2ad^-2bc-^2bd+2cd. 

146.  The  method  here  used  applies  to  any  polynomial, 
no  matter  how  many  terms  there  may  be. 

Suppose  the  polynomial  written  down  twice,  the 
second  time  immediately  below  the  first,  and  call  the  first 
line  the  multiplicand  and  the  second  line  the  multiplier, 
and  let  us  see  what  terms  make  up  the  product,  which  of 
course  is  the  square  of  the  given  polynomial. 

First y  it  is  plain  that  the  square  of  each  term  of  the 
polynomial  must  be  part  of  the  product  required. 

Second,  each  term  of  the  multiplicand  being  multiplied 
by  each  term  of  the  multiplier,  it  will  be  evident,  if  looked 


172  POWERS   AND   ROOTS. 

at  carefully,  that  another  part  of  the  product  is  made  up 
of  twice  the  product  of  each  term  of  the  -polynomial  by 
each  term  that  follows  it. 

For,  consider  how  any  one  of  these  latter  described 
terms  can  arise.  Consider,  for  example,  the  term  bd  in 
the  case  worked  out  of  a  polynomial  of  four  terms.  The 
only  way  a  term  bd  can  occur  in  the  product  is,  first,  for 
d  of  the  multiplicand  to  be  multiplied  by  b  of  the  multi- 
plier, and,  second,  for  b  of  the  multiplicand  to  be  mul- 
tiplied by  d  of  the  multiplier,  and  the  two  terms  thus 
produced  being  combined  together  we  obtain  2bd  as  one 
term  of  the  product  required.  Notice  that  d  is  one  of  the 
letters  \}oi2X  follow  b  in  the  polynomial  considered. 

The  reason  that  we  have  no  term  db  in  the  product  is 
because  that  term  has  alread}^  been  accounted  for  by  con- 
sidering it  the  same  as  bd. 

Of  course,  what  is  here  said  about  bd  will  apply  to  any 
two  different  terms  of  the  given  polynomial. 

147.  Thus,  the  square  of  any  polynomial  may  be 
written  down  directly  b)^  writing  first  the  sum  of  the 
squares  of  each  of  the  terms  of  the  given  polynomial,  and  to 
this  S2im  adding  twice  the  product  of  each  term  by  each  term 
that  follows  it  i7i  the  given  polynomial. 

148.  In  applying  this  method  it  must  be  remembered 
that  the  letters  we  have  used  may  stand  for  negative  as 
well  as  positive  numbers,  and  in  those  terms  which  are 
made  up  of  twice  the  product  of  each  term  by  each  term 
that  follows  it,  the  rule  of  signs  must  be  observed. 

By  the  method  just  explained  the  square  of  a—x—'Z 
would  be         a"^ -\-x'^ -{-z'^—^ax—^az-\-'lxz, 
the  last  term  having  the  sign  -f-  because  the  two  terms 
multiplied  together  to  produce  this  term  have  like  signs. 


ROOTS   OF   MONOMIALS.  1 73 

Write   down   the    square   of   each   of   the   following 
polynomials: 

1.  a-\-b—c.  15.  a-\-b-\-c—d—e. 

2.  r+s—x.  16.  2a—2b—c—d—e. 

3.  m-\-r—x-\-y,  17.  x-\-y+2—a—b—c. 

4.  a  +  2b—c—d.  18.  u-\-2v+Zw~4:X. 

5.  a—2b—Zc+d.  19.  w2  4_2z;2  4_a;_|-^2 

6.  2a—2n  +  ^x—y,  20.   ^^2  4.(^2z/)2-f  (3ze')2— j»r. 

7.  ;t:4-jJ^— 2z^— 3z/.  21.  m  +  2r—bs-\-t, 

8.  j»;2+;tr+l.  22.  ;r>/-f-a<^— ^2_|_^2^ 

9.  x^-^2x+l.  23.   a^<:— .ry^+;t:2-f_y2^ 

10.  ;r^— Sjt^H-Sjr— 1.  24.  mrs'^  —  uv-{-st—w, 

11.  ;i:3+3;i:2j^4-3ji:j/2+y.  25.  («^)2-;t2j/  +  (2x)2-j-jj;. 

12.  ;»;2+jj/2— 2'2.  26.  «  +  2jr+«;tr+2^;t:2. 

13.  a^— ;i:^— jK^.'  27.  m-\-s-\-xyz—ti'^v. 

14.  r2--«rH-52— ^.y.  28.  «<^;i:2— «z;^2^^^^)2^ 

EXERCISE  81. 

Roots  of  Monomials. 

149.  We  have  learned  how  to  find  any  power  of  a 
monomial,  the  square  and  cube  of  a  binomial,  and  the 
square  of  any  polynomial.  The  reverse  process  of  going 
back  to  the  number  or  expression,  when  the  power  is 
given,  is  called  Evolution. 

150.  The  number  or  expression  found  by  evolution  is 
called  a  Root  of  the  number  or  expression  given. 

Note  that  this  is  an  entirely  different  use  of  the  word  root  from  that 
of  Art.  103. 

151.  As  there  are  square,  cube,  4th,  5th,  etc.,  powers, 
so  there  are  square,  cube,  4th,  5th,  etc.,  roots. 


174  POWERS    AND    ROOTS. 

The  square  root  of  a  given  expression  means  that  ex- 
pression which  squared  ^\W  produce  the  given  expression. 

The  cube  root  of  a  given  expression  means  that  ex- 
pression which  cubed  will  produce  the  given  expression. 

The  4th  root  of  a  given  expression  means  that  expres- 
sion which  raised  to  the  fourth  power  will  produce  the 
given  expression,  etc. 

For  example,  2^  =  8;  therefore,  the  cube  root  of  8  is  2, 
24  =  16;  therefore,  the  fourth  root  of  16  is  2,  etc. 

152.  A  root  is  indicated  by  the  sign  i/,  called  a 
Radical  Sign.  A  horizontal  line  usualh^  extends  from 
the  upper  end  of  the  radical  sign  over  the  expression  of 
which  the  root  is  to  be  extracted.     See  Art.  80. 

To  indicate  what  root  is  to  be  extracted,  a  small 
figure,  called  the  Index  of  the  root,  is  placed  in  the  angle 
of  the  radical,  excepting  in  case  of  the  square  root,  in 
which  the  index  is  almost  never  used. 

Thus,  the  square  root  of  16  is  indicated  by  1^16,  the 
cube  root  of  8  is  indicated  by  #  8,  the  fourth  root  of  16 
is  indicated  by  1/16. 

A  letter  may  be  used  as  the  index  of  a  root.  Thus, 
V a  means  the  nWi  root  of  a,  that  is  a  number  which 
raised  to  the  7i  th  power  will  produce  a. 

153.  We  must  notice  one  important  distinction  be- 
tween raising  to  a  power  and  extracting  a  root.  If  we 
have  an  expression  given  to  be  raised  to  a  given  power, 
we  obtain  only  one  result,  but  if  we  have  an  expression 
given  to  extract  a  given  root  we  may  sometimes  obtain 
more  than  one  result. 

For  example,  5^  =  25,  hence  we  say  l/25=5;  but  also 
(—5)2  =  25;  hence  we  say  l/25=— 5. 


ROOTS    OF    MONOMIALS.  1/5 

It  appears  thus  that  there  are  two  numbers  +5  and  —5, 
either  of  which  is  a  square  root  of  25.  The  two  results 
are  often  written  together  by  means  of  the  double  sign  it. 
Thus,  1/25= ±5. 

1.  What  are  the  two  square  roots  of  100? 

2.  What  does  2^  equal? 

3.  What  does  (—2)'*  equal? 

4.  Are  there  two  4th  roots  of  16  ? 

5.  Are  there  two  4th  roots  of  81  ? 

6.  If  n  stands  for  an  even  n amber  is  there  more  than 
one  n  th  root  of  a  given  number  ?     Why  ? 

7.  A  sixth  root  of  64  is  2,  give  another  number  that  is 
also  a  6th  root  of  64. 

8.  Any  even  root  of  a  positive  number  may  be  either 
positive  or  negative.     Why  ? 

164.  If  any  number  be  raised  to  any  even  power,  that 
number  is  used  an  even  number  of  times  as  a  factor,  and 
therefore  the  result  must  be  positive  w^iether  the  number 
given  was  positive  or  negative.  In  other  words,  there  is 
no  positive  or  negative  number  which  raised  to  an  even 
power  will  give  a  negative  result,  so  we  cannot  find  an 
even  root  of  a  negative  number. 

155.  If  an  expression  be  raised  to  an  odd  power,  that 
number  is  used  an  odd  number  of  times  as  a  factor,  and 
therefore  the  result  is  an  expression  of  the  same  sign  as 
the  one  given.  Therefore,  any  odd  root  of  an  expression 
has  the  same  sign  as  the  expression  itself. 


176  POWERS   AND    ROOTS. 

156.  To  find  any  root  of  any  expression  we  naturally 
look  to  see  how  the  corresponding  power  was  obtained, 
and  then  go  through  the  work  backward  if  possible, 
thus  returning  to  the  expression  from  which  we  started 
in  the  case  of  involution. 

We  have  found  that  {cCy^a"''  that  is,  «"  is  an  expres- 
sion which  raised  to  the  r  th  power  gives  a'"";  therefore. 

Hence,  to  extract  the  n  th  root  of  a  power  of  an  expres- 
sion, we  divide  the  exponent  of  the  given  power  by  the 
index  of  the  root,  but  in  order  to  perform  the  division, 
the  exponent  of  the  power  must  be  a  multiple  of  the 
index  of  the  root. 

We  cannot  extract  the  square  root  of  «^,  because  5,  the 
exponent  of  the  power,  is  not  a  multiple  of  2,  the  index 
of  the  root. 

157.  To  find  the  n  th  root  of  the  product  of  two  factors. 
We  know  that  a"b"={aby\ 

therefore,  V  a''b''-==-i/\aby'=ab. 

In  this  result  the  first  factor,  a,  may  be  found  by  taking 
the  n  th  root  of  a'\  the  first  factor  of  the  given  expression, 
and  the  second  factor,  b,  of  the  result  may  be  found  by 
taking  the  n  th  root  of  b'\  the  second  factor  of  the  given 
expression.  Therefore,  the  n'Oa  root  of  the  product  of 
two  factors  is  found  by  taking  the  product  of  the  n  th 
roots  of  those  factors. 

158.  Of  course  the  same  argument  may  be  used  with 
more  than  two  factors,  and  hence,  evidently,  the  n  th  root 
of  the  product  of  several  factors  is  fotmd  by  taking  the 
product  of  the  n  th  roots  of  those  factors. 


EXAMPLES.  177 

169.  To  find  the  nth.  root  of  the  quotient  of  two 
expressions. 

We  know  that  ^""(^j 

therefore  ^|,=  ^g)   =^ 

In  this  result  the  numerator,  a,  is  found  by  taking  the 
n  th  root  of  the  given  numerator,  and  the  denominator,  d, 
is  found  by  taking  the  71  th  root  of  the  given  denominator. 
Therefore,  ^^e  nth  root  of  the  quotieyit  of  two  expressio7is 
is  found  by  taking  the  quotient  of  the  n  th  roots  of  those 
expressions. 

EXERCISE  82. 

Examples. 

Find  the  square  root  of  each  of  the  following  twelve 

expressions  : 

^2^2  4 

5. 


I. 

a'^x". 

2. 

9a6;t2. 

9^2 

3- 

16^*' 

4. 

r^^V*. 

o    10 ; — 

•    49«2;t:2*  •  y^      zX 

49j;2y4  ^  ^ 


.T-^ 


8.  '-^~.  12.  wV*-^— . 

Find  the  cube  root  of  each  of  the  following  ten  ex- 
pressions : 

13.  a^xK  15.  -|^.  17.  3sZT.-- 

14.  8«»^«^^  16.   -64a»;»;i2^        18,   77^  Ti"- 
12 


178  POWERS   AND    ROOTS. 

19. 


27     x^ 

21. 

^6,j,6^6 

-8 

-Sx'  64' 

U'^V'^W^ 

•  x'-y^' 

-64a^'d^---Sa\ 

22. 

a^x^ 

r^r« 

20. 

23.  Find  the  fourth  root  of  ^^^^^r*. 

24.  Find  the  square  root  of  X'oa^x^y^'^ ,  and  then  the 
square  root  of  this  result. 

25.  Find  the  fourth  root  of  IQa^x'^j/'^^. 

26.  Find  the  cube  root  of  a'^  ^ d'^  ^ c'^  "^ ,  and  then  the 
fourth  root  of  this  result. 

27.  Find  the  sixth  root  of  a^^d^'^c^*,  and  then  the 
square  root  of  this  result. 

28.  Find  the  twelfth  root  of  ^'^^^^  V^*. 

29.  Find  the  square  root  of  <2^^<^^2^24^  then  the  cube 
root  of  the  result,  and  then  the  square  root  of  this  second 
result. 

30.  Find  the  fifth  root  of  —32^5^-1  ^y\ 

31.  Find  the  seventh  root  of  128^7^7^1*. 


X       V 

32.   Find  the  square  root  of  '    "  ^   ^    -. 

160.  Before  leaving  the  subject  of  the  roots  of  mono- 
mials, it  is  well  to  notice  that  what  we  have  learned 
may  be  used  to  find  the  roots  of  arithmetic  numbers, 
when  the  numbers  given  have  exact  roots. 

We  resolve  the  number  into  its  prime  factors,  and  ex- 
press it  as  the  product  of  various  powers  of  these  prime 
factors,  then  divide  each  exponent  by  the  index  of  the 
required  root.  When  the  resulting  factors  are  multiplied 
together  the  required  root  is  found. 


SQUARE    ROOT   OF    POLYNOMIALS.  1 79 

Suppose  we  wish  to  find  the  square  root  of  53361. 


3 

53361 

3 

17787 

7 

5929 

7 

847 

11 

121 

11 

Hence,  53361=3^  x  7^  x  11^; 

therefore,  1/53361  =  3  x  7  X  11=231. 

33.  Find  the  square  root  of  5184. 

34.  Find  the  square  root  of  43204. 

35.  Find  the  cube  root  of  85184. 

36.  Find  the  cube  root  of  32768. 

EXERCISE  83. 

Square  Root  of  Polynomials. 

161.  To  find  out  how  to  extract  the  square  root  of  a 
polynomial  we  must  see  how  the  polynomial  was  pro- 
duced b}^  squaring.     We  know  that 

Therefore,  w^e  know  that  the  square  root  o{  x'^ -\-2xy-\-y^ 
is  x-}-y. 

Our  problem  then  is  this:  Given  the  expression, 
x'^-^2xv-\-j'^,  to  find  from  it  the  expression,  x+j. 

The  first  term,  x,  of  the  root  is  the  square  root  of  the 
first  term,  x'^ ,  of  the  given  expression. 

Let  us  set  down  this  term,  x,  already  found,  and  sub- 
tract its  square  from  the  given  expression.  There  re- 
mains of  the  given  expression  2xj/-{-y'^  or  (2x-\-y)j/. 

From  this  we  see  that  the  second  term,  j',  of  the  root 
will  be  the  quotient  when  the  remainder  just  found  is 
divided  by  2x-{-j/. 


l80  POWERS   AND    ROOTS. 

This  divisor,  2x-{-y,  consists  of  two  terms,  the  first  of 
which  is  t^ice  the  portion  of  the  root  already  found,  and 
the  second  is  the  new  term,  jy,  itself. 

The  work  may  be  arranged  as  follows: 

Given  Expression.        Root. 


'I'X.-^y 


2xy-^jy^ 
1xy-\-y'^ 


After  the  first  term  x,  of  the  root  has  been  found,  its 
double,  2x,  is  used  as  a  trial  divisor  by  which  to  divide 
the  remainder,  ^xy+y"^.  We  see  that  the  first  term,  2xy, 
of  this  remainder  when  divided  by  the  trial  divisor,  2x, 
gives  y,  from  which  we  judge  that  y  is  the  next  term  oi 
the  root.  When  the  y  is  thus  found  it  is  added  to  the 
trial  divisor,  2;r,  giving  the  complete  divisor,  2x+y,  and 
this  is  multiplied  by  ^,  giving  the  expression,  2xy-\-y'^. 

162.  Of  course  we  may  obtain  by  this  process  the 
difference  of  two  numbers  for  our  square  root  as  well  as 
the  sum,  as  in  the  example  just  given.  This  will  be 
plain  by  working  out  another  example. 

To  find  the  square  root  oi  \a'^'-\2ab-\-^b'^ , 

Arrange  the  work  thus  : 

4^2 

4.a-Zb 


-12ab+db^ 
-12ab+db^ 


Here  the  first  term  of  the  remainder,  —12ab,  when 
divided  by  the  trial  divisor,  ia,  gives  the  quotient,  —3^. 
Hence  we  judge  that  —Sb  is  the  next  term  of  the  root, 
and  upon  trial  this  proves  to  be  right. 


SQUARE  ROOT  OF  POLYNOMIALS.       l8l 

Find  the  square  root  of  each  of  the  following  ex- 
pressions : 

1.  «2+4«<^+4^^  II.  x'^—^x-^+x*. 

2.  4a''-iab+b\  12.  dx^-lSx"- +d. 

3.  da'^-lSab+^d^.  13.  a^-2a^x^-\-x^. 

4.  4a2_16«-fl6.  14.  aH^-h2adcd-{-c^d'^. 

5.  a^—2a'^d'^-{-d*.  15.  a^x'^  —  Iadx^  +  d^x'^, 

6.  a^-  —  2ad-^-\-d\  16.  4x^—4nx'^j'-{-?i^j\ 

7.  ;t-4-f22_>'2_^y.  17.    9a4<^2_1^^3^3+9^2^4^ 

8.  .r*  +  2;r2  +  l.  18.  «4^'*-6«2^2^9. 

9.  X^-h2x^+x'^.  19.    «*/^6_2^2^3^5_^^10^ 

10.  ;»;«— 2;t-*H-:r2.  20.   «2^2jr2_>'2— Sa^^r^r^j^+lGr^jtr^ 

163.  Thus  far  the  polynomials  of  which  we  have  ex- 
tracted the  square  root  have  been  in  ever\'  case  those  of 
three  terms. 

The  above  process,  however,  can  be  extended  so  as  to 
find  the  square  root  of  any  polynomial  which  is  a  perfect 
square,  no  matter  how  many  terms  the  polynomial  con- 
tains.    For  example,  to  find  the  square  root  o^ 
a^-^d'^-hc'--{-2ad-}-2ac+2dr. 

First  arrange  the  expression  according  to  powers  of 
some  letter,  say  a,  and  write 

a^  +2ad-{-2ac+  d'^  +2dc-^c'^ . 

The  first  term,  a^,  of  this  polynomial  is  produced  bj^ 
squaring  a.  Therefore,  the  first  term  of  the  root  is  a, 
and  the  wko/e  root  is  «  +  something,  and  this  something 
is  what  we  wish  to  find. 

Proceeding  as  before  with  the  first  term  of  the  root,  a 
part  of  the  process  may  be  arranged  thus  : 

a'-  -^2ad+2ac-^d'' -\-2bc-\-c''  (  a 


2ah-^2ac-^b'^-\-2bc+c'^ 


1 82  POWERS   AND    ROOTS. 

Now  twice  a  used  as  a  trial  divisor  would  suggest  b  for 
the  next  term  of  the  root. 

Call  the  next  term  b  and  proceed  as  before,  and  the 
work  would  stand  thus: 

a"^ ■^lab^lac^b'' ^-Ibc^-c''  (  a^b 

"la^-b 


2ab-^-2ac+b'^+2bc+c'^ 
lab  +^2 


lac        -\-2bc-i-c^ 

There  is  sfzll  a  remainder,  so  we  have  not  yet  found 
the  entire  root,  but  the  root  is  a-\-b-^  something,  and 
this  something  is  what  we  wish  to  find. 

Now  let  us  consider  a-j-b,  the  part  of  the  root  already 
found,  as  a  single  term,  and  use  it  as  we  have  before  used 
the  first  term  of  the  root.  We  must  then  take  twice 
a-hb  and  use  it  as  a  trial  divisor  by  which  to  divide  the 
last  remainder,  2ac-\-2bc-\-c'^,  from  which  we  judge  that 
c  is  the  next  term  of  the  root.  When  the  c  is  thus  found 
it  is  added  to  the  trial  divisor,  2a-\-b,  giving  the  complete 
divisor,  2a-\-2b-i-c,  and  t/iis  is  multiplied  by  c,  giving  the 
expression  2ac-\- 2bc+  c"- . 

The  work  from  the  beginning  will  now  stand  thus: 

a"" -^2ab-\-2ac-\-b'' ^■2bc-^c''  (  a-^b^c 
a"" 

2a  +  b 


2ab+2ac+b^-\-2bc+c'^ 
2ab  4-/^2 


2a+2b-\-c 


2ac         -\-2bc-\-c^ 
2ac         ^-2bc^c'^ 


As  there  is  now  no  remainder  the  process  is  ended  and 
the  root  \s,  a-\-b-\-c.  If  after  finding  the  third  term  of 
the  root  there  were  still  a  remainder,  we  would  group  the 
three  terms  thus  found  into  a  single  term,  and  use  this 
group  as  we  have  always  used  the  first  term  of  the  root. 


CUBE   ROOT   OF*'   POLYNOMIALS.  1 83 

The  process  may  evidently  be  extended  to  finding  the 
square  root  of  any  polynomial  that  is  a  perfect  square, 
no  matter  how  many  terms  the  polynomial  may  contain. 

164.  From  what  has  been  said  we  see  that  the  Method 
of  finding  the  Square  Root  of  any  Polynomial  is  as 
follows  : 

/.  Arrajige  the  terms  according  to  the  powers  of  some 
letter. 

II.  Find  the  squa7'e  root  of  the  first  term,  zvrite  it  as  the 
first  term  of  the  root,  and  siibtraci  its  square  from  the  give?i 
expression. 

III.  Use  twice  the  portion  of  the  root  already  found  as  a 
trial  divisor,  and  divide  the  remainder  just  found  by  this 
trial  divisor.  Add  the  quotient  to  the  root  and  also  to  the 
tria  divisor. 

IV.  Multipiy  the  complete  divisor  by  the  term  of  the  root 
last  obtained  and  subtract  the  product  from  the  rej?tainder. 

V.  Repeat  III  and  IV  until  there  is  no  remainder. 

Find  the  square  root  of  each  of  the  following  poly- 
nomials : 

11.  x'^-\-'^xy-\-2xz-\-\y'^-\-A:yz-\-z'^ . 

12.  ;»;26jrj/— 4;i:^+9>'2^12j/^-f4^^ 

13.  4.r2^16.rj/— 4x^-f  l(y/2— 8j'^H-^2^ 

14.  ^•*-2a2<^+2a2r-f^-_2^^_}_^2^ 

15.  4«4-12«2^2_g^2^_f_9^4_^12^V+4^2^ 

16.  «4+2«2^2_^2«V2-f^*+2<^2^2_j.^4^ 

17.  a-+2«^  +  2a<:+2a^+^2^2^^+2^i/+^2_^2r^4-^^ 

18.  a"-  ^^ab—^ac^-W" ^-Vlbc-^Sic"- . 

19.  a«-f-2a5-}-3a*-f4a'^4-3a2  +  2^  +  l. 


1 84  POWERS   AND    ROOTS. 

20.  9x*  +  lHa^x'^-{-6x'^-\-9a^-i-Qa^-\-l, 

"V^  -y^  'Y*'"  'V* 

21.  ^  +  2^+S~  +  2--hl. 
a*        a^        a^        a 

22.  ji:8  4- 2;t:«+ 3:^4 +  2.^2  +  1. 

23.  a^b^+2a''b''cd-4:a''b'^e+c'^d'^-Acde-\-Ae'^, 

24.  aH'^-^2a^bc'^d+2a^be-\-c'^d'^'\-2c'^de+e'^. 

25.  aH''c''-]-2abcde+2abcf^-d''e''+2def^p, 

EXERCISE  84. 

Cube  Root  of  Polynomials. 

165.  To  find  how  to  extract  the  cube  root  of  a  poly- 
nomial we  must  see  what  the  cube  of  an  expression  is 
and  then  return,  if  possible,  from  the  cube  back  to  the 
expression  from  which  this  cube  was  obtained. 

166.  We  know  that 

{x+yy=x^-^Zx'^y  +  2>xy'^-\-y^. 
Our  problem,  then,  is  this :     Given  the  expression, 
x^-{-Sx'^y-\-Sxy'^-{-y^,    to   find   from   it   the   expression, 
x-^y,  which  is  the  cube  root  of  the  given  expression. 

167.  We  see  that  the  first  term,  x,  of  the  root  is  the 
cube  root  of  :r^,  the  first  term  of  the  given  expression. 

Set  down  the  x  as  the  first  term  of  the  root  and  sub- 
tract its  cube  from  the  given  expression,  and  we  have  a 
remainder, 

Sx^y+Sxy'^+y^,  or  (Sx^-\-Sxy-{-y^)y, 

We  see  from  this  that  the  second  term,  y,  of  the  root 
is  the  quotient  obtained  by  dividing  this  remainder  by 
Sx^  +  Sxy+y'^. 

Now  this  divisor  is  composed  of  three  terms,  of  which 
the  first  is  8  times  the  square  of  the  first  term  of  the  root. 


CUBE    ROOT    OF    POLYNOMIALS.  1 85 

the  second  is  3  times  the  product  of  the  first  term  of  the 
root  and  the  new  term,  y,  and  the  third  is  the  square  of 
the  new  term  of  the  root. 

The  sum  of  these  three  terms  constitute  the  complete 
divisor,  while  the  remainder  found  b}^  subtracting  x-^  from 
the  given  expression  is  the  complete  dividend. 

This  complete  divisor  contains  two  terms  which  involve 
the  jK,  which  is  not  yet  supposed  to  be  known.  However, 
we  may  get  something  of  an  idea  of  what  the  second  term 
of  the  root  must  be  by  using  the  ^rsf  term  of  the  above 
remainder  as  a  trial  dividend  and  3  times  the  square  of 
the  first  term  of  the  root  as  a  trial  divisor,  and  then  if  the 
number  we  get  by  this  division  is  correct  the  complete 
divisor  (which  can  then  be  found)  when  multiplied  by 
the  new  term  of  the  root  must  give  the  complete  divi- 
dend, i.  e.,  the  remainder. 

168.  The  work  may  be  arranged  as  follows  : 


Zx'^^-Zxy^y'' 


3.r2_>/-J-3;r|/2-hj^/3 
Sx'^y-^Sxy^-\-y^ 


169.  The  remark  made  under  square  root  (Art.  162) 
about  the  —  sign  applies  here  as  well,  as  is  illustrated  in 
the  following  example  : 

Find  the  cube  root  of  x^—Qx'^y+12xy'^ -\-y^. 

The  work  is  as  follows  : 


X 

Sx''-Qxy+4y^~ 


x^  —  6x'^y-{-12xy-—Sy^  (  x—2y 
3 


—(dx'^y+12xy—Sy^ 
-6x''y+12xy''-Sy^ 


1 86  POWERS    AND    ROOTS. 

Find  the  cube  root  of  the  following  expressions  : 

3.  x^v^  +  12x-v'^-j-4Sxy^+64y\ 

4.  Sa^  —  60a'^x-\-loOax'^  —  12DX^. 

5.  27.r«  — 54x*r2'  +  36x2j,/2^2_3^3^3^ 

6.  8jr«  +  12a'4_^6;t:2  +  l. 

7.  a^x^-SaHx^'y^+Sad^xy^-Py. 

8.  8a•'^^^-24a2^V+24«^^•^-8^^ 


9-  ^+3^+3x24-^^ 


10.  4+34+3^+^. 

a^       a^b         b-^       b^ 

170.  So  far  all  the  expressions  of  which  the  cube  root 
was  required  were  polynomials  of  four  terms,  but  we  ma}^ 
have  a  polynomial  of  more  than  four  terms  of  which  the 
cube  root  is  required.  In  this  case  the  process  already 
given  may  be  extended  as  in  the  case  of  the  square  root, 
viz.:  Find  two  terms  of  the  root,  as  already  explained, 
and  then  consider  these  two  terms  as  a  single  term  and 
use  their  sum  the  same  as  a  single  term  was  used  before. 

Find  the  cube  root  of  each  oi  the  following  three  ex- 
pressions : 

11.  ««+3«-5+6«4  +  7^3_{_(3^2_^3^^1_ 

12.  Ji;6— 6;t:^4-9,r4  +  4;f»-9jf2  — 6.r— 1. 

13.  G4.r«  +  192.;r-^  +  144;r^-32.T=^-36.r^  +  12.r-l 


CHAPTER  XIII. 

HARDER  FACTORS,   MULTIPLES, 
AND  FRACTIONS. 

EXERCISE  85. 

Factors  Common  to  all  the  Terms  of  an  Expression. 

171.  In  Chapter  IX  the  subject  of  factors,  multiples, 
and  fractions  was  treated  to  some  extent,  but  the  work 
there  was  confined  to  the  case  of  monomials.  We  now 
resume  the  same  subject,  but  treat  of  more  complicated 
expressions  than  before. 

172.  The  simplest  case  of  factors  of  pol3momials  is 
where  the  same  factor  is  seen  to  be  common  to  all  the 
terms  of  the  polynomial.  In  this  case  the  polynomial  nia}^ 
be  written  in  a  simpler  form,  by  dividing  each  term  by  this 
common  factor,  enclosing  the  quotient  in  a  parenthesis, 
and  writing  the  common  factor  outside  the  parenthesis  as 
a  multiplier. 

Examples. 

1.  Factor  ax'^-\-ax-{-a. 

Here  ti  is  seen  to  be  a  common  factor  of  each  term  ;  therefore,  re- 
moving this  factor  we  have 

(7X--\-llX-\-a  =  (7{x'-\-X-]-l.  ) 

2.  Factor  oax^  •i-loax'^  -j-20ax-\-50a. 

Here  5^'  is  seen  to  be  a  factor  of  each  term;  therefore,  removing  this 
factor,  we  have 

Find  the  factors  of  each  of  the  following  expressions  : 

3.  x'^-hx'^-j-x^.  5.  a-dc-ha-dd'^+a'^de. 

4.  m'^x'^-{-n'^x^-\-r^x'^.  6.   rsLr'^-\-rsf-j/'^-\-rs'^^^2'^. 


1 88  HARDER    FACTORS,   MULTIPLES,  ETC. 

7,   auv^ -\-buvx-\-uvw. 

10.  10^2^2  _]5^^3_25^^2^ 

11.  ^m"^ x"^ —hm'^y'^  —^m"^ x'^  —  Ihm'^y'^. 

12.  Z?>r'^x^-^bbr'^x^y—mr'^x'^y^. 

173.  Sometimes  07ie  factor  is  common  to  some  of  the 
terms  of  an  expression,  and  aiiother  factor  common  to 
other  terms.  In  this  case  you  can  so^netifnes,  though  not 
always,  simplify  the  expression  by  taking  out  the  common 
factors  where  they  can  be  taken  out. 

13.  Factor  ax-\-ay—a2-{-bx-\-by—b2. 

Here  the  first  three  terms  have  a  common  factor  a,  and  the  last 
three  terms  have  a  common  factor  h.  Taking  out  a  from  the  first 
three  terms  and  b  from  the  last  three  terms,  we  may  write 

ax-\-ay  —  az-\-bx-\-by—bz=a{x-\-y—z)-\-b{x-\-y—z.) 
Now  it  is  plain  that  in  this  expression  the  factor  {x-\-y—z)  is  com- 
mon.    Hence,  we  may  write 

a{x^y-z)+b[x-^y-z)  =  [a+b)[x^y~z). 
Therefore,  putting  the  expression  we  started  with  equal  to  this  last, 
we  get  ax-\-ay —az-\-bx-\-by  —  bz  =  [a-\-b){x-\-y —z). 

Factor  each  of  the  following  expressions  : 

14.  ax  +  bx-\-cx-\-2ay-\-2by-\-2cy. 

15.  ax+2ay-\-Zaz—2bx—Aby—^bz. 

16.  ax'^y-\-bx'^2-\-cx'^u-\-axy^bx2-^cxii. 

17.  x^y'^—x^y-{-x^-^y'^—y-\-l. 

18.  abxy2-\-abx2iv-{-abxu'-\-aby2-\-ab2iv-{-abw. 

19.  abxyz"^  -{-abxy2-^abxy-\-abx2'^  +abx2-{-abx  . 

20.  a7ivwx-^buvzi>x-j-auvwy-{-b2n'wy, 

21.  xy -\- ax -\- ay -\- a'^ .  v 

22.  xy — ay — bx-\-ab. 

23.  x'^y—ay—1b'^x'^-\-1ab'^. 


FORMATION    OF    CERTAIN    PRODUCTS.  1 89 

EXERCISE  86. 

Formation  of  Certain  Products. 

174.  In  order  readily  to  factor  expressions,  we  must 
know  what  expressions  were  multiplied  together  to  pro- 
duce the  expression  which  is  given  us  to  factor.  This 
knowledge  is  gained  by  practice  in  writing  down  the  pro- 
ducts of  a  few  forms  of  expressions.  Those  we  take  are 
binomial  factors.  We,  have  already  found  out  in  the 
chapter  on  multiplication  how  to  multiply  two  binomials 
together,  but  it  is  very  important  that  the  student  should 
be  able  to  write  down  rapidly  certain  products  by  inspec- 
tion, that  is,  without  actually  going  through  the  work  of 
multiplication. 

175.  The  first  case  to  consider  is  that  of  the  product 
of  the  sum  of  two  numbers  by  the  difference  of  those 
numbers. 

Thus,  suppose  it  is  required  to  write  out  the  product 
{a-^b){a—b~).     By  multiplication  we  find  that 

{a^-b){a-b)^a''-b''. 
That  is,  the  differe^ice  of  the  squares  of  a  and  b. 

Now  as  a  and  b  may  stand  for  any  numbers  whatever, 
we  may  state  that 

The  siini  of  any  two  7iumbers  niiiltiplied  by  the  difference 
of  those  numbers  is  equal  to  the  difference  of  the  squares  of 
those  numbers. 

Examples. 

Write  down  by  inspection  each  of  the  following  products: 

1.  {x  \-a'){x—a).  4.   {fn-\-n~){7n—7i). 

2.  {x^-\){x-\).  5.   (a^  +  bXa-'^^b). 

3.  (jc-^2y)(^x-2y).  6.   (^2+4)(^2_4), 


190  HARDER    FACTORS,   MULTIPLES,   ETC. 

9.   (x2_^4)(;r2-4). 
10.   {x^—2ab){x^+1ab). 

176.  The  second  case  of  the  multiplication  of  two 
binomials  is  that  where  the  first  term  is  the  same  in 
each  binomial. 

Thus,    suppose  it  required  to. write  out  the  product 
{x-\-a){x-^b).  By  actual  multiplication  we  could  find  that 
(x-\-a){x-\-b)=x'^-{-ax-\-bx-\-ab 
=x'^-{-(a-\-b)x-j-ab. 

Notice  that  the  first  term  of  the  product  is  x'^,  the 
second  term  is  x  with  a  coefficient  equal  to  the  sum  of 
the  second  terms  of  the  given  binomials,  and  the  third 
term  of  the  product  is  the  product  of  the  second  terms  of 
the  given  binomials. 

As  X,  a,  and  b  are  not  in  any  way  restricted,  but  may 
stand  for  any  numbers  whatever,  the  result  reached  is 
perfectly  general,  and  therefore 

The  p7^o  duct  of  any  tzvo  bi7iomials  in  which  the  first  terms 
are  alike,  is  eqnal  to  the  square  of  the  first  term,  plus  the 
first  term  with  a  coefficient  equal  to  the  sum  of  the  seco7id 
terjns,  phis  the  p7vduct  of  the  second  terms. 

Of  course,  due  attention  must  be  paid  to  the  signs  of 
the  terms  in  writing  out  such  products.     Thus, 
(:t--5)Cr  +  2)=.r2  4.(_5^2Xr+(-5x2) 
=  ;t-2-3jt:-10. 

Write  down  the  products  in  each  of  the  following : 

11.  (x+«)(x+2).  14.   (x-f-5)(x+6). 

12.  (^-2)(^^+l).  15.   (^--5)(x-6). 

13.  (:f+3)(jr-5).  16.   (;r+2)(x-15). 


EXPRESSIONS    OF    THE    FORM    x^  —  a^.  191 

17.  (;^4-l)(-r+5).  34.  (^-6)(;t-+5). 

18.  {x-\){x-y).  35.  {x-^1a){x-Vlb). 

19.  (;f+l)(;r-5).  36.  (;r-8^)(;r+^2)^ 

20.  (jr-f2)(.r+4).  37.  (x-«2)^^_^^2>)^ 

21.  (^-l)(^+7).  38.  (;t:2-l)(.r2  +  2). 

22.  (jr+10)(.r— 11).  39.  (a  +  ^^)(«— r^). 

23.  (j>;+25)(.r— 4).  40.  {a—?nr^)(a—7ris'^). 

24.  (a-— ^)(.i-  +  8).  41.  (jt:+/;2;/)(jt;  +  wj/). 

25.  Cr-10)(;i--10).  42.  i^x'^-a^-y'^Xx'^-ay). 

26.  (:r+10)(-r— 5).  43.  (2.a-b')(2a+Zb). 

27.  (a'-+5)(;»;+12).     '  44.  (^2a-b^){2a-c^), 

28.  (a'-4)(jtr-lo).  45.  (;«;  +  3)'^)(;r-2«z'). 

29.  (x+8)(:t-  +  20).  46.     (;t:2+«2)(^2_^l)_ 

30.  (A'-f2)(^+80).  47.  a'2  +  2^^)(;»;2-3«j/).) 

31.  (jr-l)(j»;-5).  48.  {ab^-irc:){ab''-2). 

32.  (jc— 2)(;f— 3).  49.  (/;^^^  +  o)(;;^«— 4). 

33.  (a-  +  2)(:ir+3).  50.  (^2^+;ir>/2)(^.2^_^_^2)^ 

EXERCISE  87. 

Expressions  of  the  Form  x~—a^. 

177.  In  this  form,  x-—a'^,  the  letters  x  and  a  can  of 
course  stand  for  any  two  numbers  whatever  ;  so  to  speak 
of  expressions  of  the  form  x"^ — a^  is  the  same  as  saying, 
an  expression  composed  of  the  difference  of  two  square 
numbers. 

178.  We  have  learned  how^  to  write  down  at  once  the 
product  of  the  sum  of  two  numbers  multiplied  bj-  the  dif- 
ference of  those  numbers,  and  we  now  take  up  the  reverse 
process  of  finding  the  two  factors  when  their  product  is 
given. 


192  HARDER    FACTORS,    MULTIPLES,   ETC. 

As  the  sum  of  two  numbers  multiplied  by  the  dif- 
ference of  those  numbers  is  equal  to  the  difference  of  the 
squares  of  those  numbers,  it  follows  that  the  difference  of 
two  numbers,  each  of  which  is  a  perfect  square,  is  equal  to 
the  sum  of  the  square  roots  of  those  numbers  multiplied 
by  the  difference  of  the  square  roots  of  those  numbers. 


Examples. 

Find  two  factors  of  each  of  the  following  expressions  : 

I.  ^2_4.                7.  a^^2_25^2 

13- 

4a^-Wa^x\ 

2.  j»;2— 9.                8.  4a2;t:2— j/4. 

14. 

64;;^8-81;^^ 

3.   :r2-l.                 9.   ^b^-^a\ 

15. 

100-25. 

4_  x'^-a''.            10.   167^2 _8Gr2;t-2 

.16. 

2500-16. 

5,    ;t:2-4y2.             II.    a-2-4^2^'^ 

17- 

121-81x8. 

6.  ^2_4^2^2       12.  a8— «6. 

18. 

625-625^2^2 

ig,  H\x^  —  Sly\  20.  49a2.r2__49^2^2^4 

The  following  expressions  require  a  factor  to  be  re- 
moved from  each  term  before  they  are  in  the  form  of  the 
difference  of  two  squares  : 

21.  5^2  _ 5^2^,4     23.    10<2  — 10^7-2 ji-2.    25.   oax'^j'—r^ax'^ . 

22.  20x^  —  20a'^.  24.  a^'x'^  —  a'^x-j^.  26.   7?ix^  —  7u?i^x^. 

27.  tV''j'-ti)-^-^1>'*-  34.  50?i'y-dS?iy. 

28.  m'Kr-'—J7t^x'y'^.  35.  2Hxy'—QHx^j. 

29.  2x^  —  d0xj^.  36.  9?ix'^  —  o()?iy'^. 

30.  27a'^—27ad'^.  37.  ax'^—ax. 

31.  x;>/^2_9Yy3^  28.  ?^z^z£/2— 4?^z''^ze/*. 

32.  x-^2^—4y'^2\  39.  24«2^6_54^2^6 

33.  4dTiy—196fiy^.  40.  50ax"y—lSaxy. 


EXPRESSIONS    OF    THE    FORM   :i;^-{-ax+d.        193 

EXERCISE  88. 

Expressions  of  the  Form  x'-^ax-\-d. 

179.  We  have  learned  how  to  write  down  the  product 
of  two  binomial  factors,  when  the  first  terms  of  the  two 
factors  were  alike,  and  found  the  ^result  was  always  a 
trinomial.  We  further  saw  the  relation  this  trinomial 
sustained  to  the  two  binomial  factors  which  were  multi- 
plied together  to  produce  it. 

180.  We  now  take  up  the  reverse  process  of  returning 
from  the  product  to  the  two  factors  which  were  multi- 
plied together  to  produce  it. 

We  can  factor  any  expression  of  the  form  x'^-\-ax-\-b  if 
we  can  find  two  numbers  whose  sum  is  a  and  whose  pro- 
duct is  b. 

It  will  assist  some  in  finding  the  two  factors,  if  the  student  will  re- 
memjDer,  that  when  the  third  term  of  the  given  expression  is  positive, 
the  second  terms  of  the  two  factors  will  have  like  signs,  and  when 
the  third  term  of  the  given  expression  is  negative,  the  second  terms 
of  the  two  factors  have  unlike  or  opposite  signs. 

Examples. 

Find  the  factors  of  each  of  the  following  expressions  : 

1.  Jtr24-8.r  +  7. 

The  first  term  of  each  binomial  factor  will,  of  course,  be  x,  and  we 
are  to  find  the  second  terms  of  the  binomial  factors.  To  do  this  we 
must  find  two  numbers  whose  sum  is  8  and  whose  product  is  7.  Obvi- 
ously 7  and  1  are  the  only  numbers  that  have  8  for  their  sum  and  7 
for  their  product.  Therefore,  we  conclude  that  the  two  factors  sought 
are  .v-j-7  and  A--f  1. 

2.  X'^—IX-Z. 

Here  we  must  find  two  numbers  whose  sum  is  —2  and  whose  pro- 
duct is  —3,  Obviously  —3  and  1  are  the  only  numbers  whose  sum  is 
—  2  and  whose  product  is  —3.  Therefore  the  factors  are  .i*— 3  and  .r-j-l. 

13 


194  HARDER    FACTORS,   MULTIPLES,   ETC. 

3.  ^2^5^+6.  15.  x^+4x-\-4. 

4.  x'^-j-5x+4.  16.  j»;2  4-12;t'+36. 

5.  ^2^8;tr+15.  17.  jt-2  +  12^-+35. 

6.  jt:2  +  llx4-28.  .  18.  ;r2-10;tr4-16. 

7.  x''+b5x-\-250.  .  19.  x^-\-2x-S. 

8.  .r2+20ji:+100.  20.  jt-2  +  13;»;+42. 

9.  jr2  +  10;»;  +  21.  21.  :i:2-4;t;--60. 

10.  ;i;2  +  ll;t:4-18.  22.  ;r2-f22;t:4-120. 

11.  x'--dx+S.  23.  x2_|_iio.r+1000. 

12.  jtr2_i(3;t:+48.  24.  ;ir2  4-52;»:+100. 

13.  ji;2^_6^^9,  25.  ;»;2-48;i:-100. 

14.  x^-+8x-\-16.  26.  ;t;2  +  15;»;+36. 

In  the  above  examples  the  first  term  is  x"^,  and  the 
second  term  is  some  multiple  of  x.  Of  course  some 
other  expressions  than  x^  and  a  multiple  of  x  might  be 
used  for  the  first  and  second  terms  respectively,  provided 
only  that  the  first  term  is  the  square  of  something,  and 
the  second  term  is  a  multiple  of  the  square  root  of  the 
first  term,  as  in  the  following  examples: 

27.  a^-x'^-4ax-12.  34.  ;i:2^2_i0;rjj/-200. 

28.  x^-Ix^'-SB.  35.  a^-17a^-^70. 

29.  r^^x^-16rx--\-m.  36.  4;t:4  +  20x2;i;2-f  36. 

30.  n^a^ -^ SW^ a'' -\- SO.  37.  ix^-^40x-j-SQ. 

31.  aV*;i;8  +  17«r2jtr4  +  16.  38.  a'^b^  —  Ua'^d'  +  ll. 

32.  w*  +  17w2  +  30.  39.  a'^x^—5axy—o0y, 

33.  n^x*+70n''x^+Q00.       40.  ^--204-f-100. 


EXPRESSIONS   OF    THE    FORM   a^ —b^ ,  195 

EXERCISE  89. 

Expressions  of  the  Form  a^—b^. 

181.  By  trial  we  find  that  a^  —  b"^  can  be  dividad  by 
a—b2A  follows  : 

a-b  )  a^-b^  {a'^-^ab^-b"-' 

a^-a'^b  

'•      a'^b-b^ 
a'^b-ab'' 


aF'-b^ 
ab^'-b^ 

Now  remembering  that  the  divisor  multiplied  by  the 
quotient  equals  the  dividend,  we  learn  from  this  division 
ihat  a-'-b^=^{a-b){a''^ab-\-b''-y, 

and  as  a  and  b  stand  for  any  numbers  whatever,  we  can 
make  the  following  statement : 

The  difference  between  afiy  tivo  numbers,  each  of  ivhich  is 
a  perfect  cube,  ca7i  be  expressed  as  the  product  of  tivo  factors, 
one  of  which  is  the  difference  between  the  aibe  roots  of  the 
^lumbers  giveyi,  and  the  other  is  the  sqtiare  of  the  cube  root 
of  the  first  nmnber  plus  the  product  of  the  cube  roots  of  the  tivo 
mimbers plus  the  square  of  the  cube  root  of  the  second  7iumber. 

Examples. 

Factor  each  of  the  following  expressions  : 

1.  ««-8. 

Here  we  have  the  difference  between  two  numbers,  each  of  which 
is  a  perfect  cube,  ^^a^  —a-  and  \'/8:=2.  Hence,  we  write  the  two 
factors  (««-2)(^*+2^2-|-4). 

2.  x^—y^.  5.  a^b^—c^d^. 

3.  x^ — a^y^ .  6.   n^r^  —  71^ r^. 

4.  x^-n-iy^^  7.  ^x^-Tia^y\ 


96  HARDER    FACTORS,   MULTIPLES,   ETC. 

8.  64:x^-125x^y^.  15.   1000-64. 

g,  Sa^x^  —  27a^x\  16.  ji^v^w'^—xy^z^. 

10.  27a^x^-27a^x\  17.   12o-aH'^c\ 

11.  x^  —  Sa^b^c\  18.   12Da^~64a^x^yK 

12.  jc6_)/9-l.  19.   1000;i:'^-64:j/3^^ 
i^.   1  —  a^jr^.  20.  Jtr^j/^ — ?/''*' 27 ^ze/^. 

14.  125-27.  21.  8a^r«7«— 27«9a-6j/\ 

^^'  1^     "U^'  ^^"  'c'^d-'      27j/s'' 

Ux^y^     u^v^  1000     27;i:'^ 


^^'      27^'^       27^3"*  ^^'    a^fy"      a 


3A3- 


Sometimes  an  expression  assumes  the  form  of  the  dif- 
ference of  two  cubes,  after  a  factor  has  been  removed 
from  each  term,  as  in  the  following  examples: 

26.  a^x'^  —  a^. 

If  we  take  out  the  factor  a'  there  will  remain  x^—a^  of  which  the 
factors  are  x—a  and  x^ -\-qx-\-a^ .  Hence,  the  factors  of  the  given  ex- 
pression are  «^,  x—a,  and  x'^ -\-nx-\-a'^ .  Hence  the  given  expression 
equals  a^[x—a)[x^-\;-aX'^^-al). 


27. 

x^  z'^  —y"^  z"^ . 

32. 

Zfrr^fi^  —  Sm^n^ 

28. 

aH^x'^-c^x'^. 

33. 

5«3_^3_40^3^3 

29. 

a'^b''^x'^y—c^x-y. 

34. 

2a'^xy^—2a^. 

30. 

r\,U'-27r'sUK 

35. 

2^2 
2«2j;6^3 

a^y^ 

31. 

27^3     ^^^'^'' 

36. 

5a^     6x^ 
b^       8>/3- 

EXPRESSIONS    OF    THE    FORM    a^  +  d^.  197 

EXERCISE  90. 

Expressions  of  the  Form  a^-\-l>^. 

182.  By  actual  division  we  find  that  a^  +  5^  can  be 
divided  by  a-}-d  as  follows: 

a-\-b  )  a^-^-b^  (^a'^-ab+b'^ 

-aH+b^ 
-aH-ab^ 


ab'^  +  b^ 
ab^  +  b^ 

Now  remembering  that  the  divisor  multiplied  by  the 
quotient  equals  the  dividend  we  learn  from  this  division 
that  a^  +  b^  =  (a-hb)(a^-ab-\-b^); 

and  as  a  and  b  may  stand  for  any  numbers  whatever,  we 
may  make  the  following  statement : 

Tke  sum  of  any  two  numbers,  each  of  which  is  a  perfect 
cube,  can  be  expressed  as  the  product  of  two  factors,  07ie  of 
which  is  the  sum  of  the  cube  roots,  of  the  7iumbers  giveyi,  and 
the  other  is  the  square  of  the  cube  root  of  the  first  number 
ttmius  the  product  of  the  cube  roots  of  the  tivo  nuynbers  plus 
the  square  of  the  cube  root  of  the  second  number. 

Examples. 

Factor  each  of  the  following  expressions  : 

I.  ««+8. 

Here  we  have  the  sum  of  two  numbers,  each  of  which  is  a  perfect 
cube  ^/a^=rt'  and  -^8=2.     Hence  we  may  write 


2. 

x^-\-y^. 

6. 

aH^-hSaHK 

3- 

a^x^  +  b^y^. 

7- 

Sx^y^-^27y^. 

4- 

7?i^  -\-x^y^. 

8. 

64;t;3^«  +  125^« 

5 

m^+a^b^. 

9- 

64x^-h8x^y\ 

HARDER    FACTORS,   MULTIPLES,   ETC. 

10.  m^-\-n'^.  15.  %a^b^x^^-^a^b^y^, 

11.  r^-{-x^.  16.  x^y^-\-Tlx'^z^. 

12.  u^v^x^^-u^v^x^,  17.  64;t-6 +  125^3^6^9. 

13.  x^y^z^-^\.  18.  1000+64. 

14.  a^b^c^-^x^y^z^.  19.  64  +  8. 


20  ^V-^' 

-  IU 

27* 

,   ,  64^5 
^4-  l  +  -27^- 

^3  ^6^,3 
•  r«  '  27  • 

+  64 
^27' 

^^'     a^     '   27^3 

26.  -^f-  +  125. 

29. 

aH''     '  64a«  ' 

^Zy.,. 

/^•^Z^^^Ziy^ 

30. 

''^-     aH^     ' 

a'-^b'^    ' 

28   ""'■'.+ 
1000a  3^ 

1000 

31- 

1000  1000 

Sometimes  an  expression  assumes  the  form  of  the  sum 
of  two  cubes,  after  a  factor  has  been  removed  from  each 
term,  as  in  the  following  examples  : 

32.  ^2-^-3 +^5  37      5^^2^9^-3^6_^40^^•2^.6^.6^ 

33.  x^'y^z'^^-a^b^z'^.  38.   2a2r6^6_^2a2. 

34.  a'^bxy^  ■^-a'^bx^z'^.  39,  zcv^x^y^  ■\-2iv'^w^ , 

35.  ax'^yz^-^-axy.  40.   128xy^ -j- 64x* z\ 
-    16;rV^  .  bAxy^z'^  ba^       ox^ 


EXERCISE  91. 

Expressions  of  the  Form  x^-{-a^x^-\-a'^. 

183.  Some  expressions  that  do  not  appear  to  come 
under  any  case  thus  far  considered  may  be  made  to  do  so 
by  adding  and  subtracting  some  expression.    Such  is  the 


EXPRESSIONS   OF   THE    FORM   ;ir* +^^^- H"^*.     199 

case  with  the  expression  we  are  dealing  with  in  this  ex- 
ercise. We  notice  that  the  expression  here  considered, 
viz.:  x'^  +  a^x'-'i-a'^,  is  ahiiost  of  the  form  of  a  perfect 
square,  viz.:  the  square  of  x'^-\-a-,  and,  indeed,  if  the 
middle  term  were  only  2a'^x^,  instead  o{  a-x'-,  the  expres- 
sion here  considered  would  be  a  perfect  square.  So  we 
make  the  expression  a  perfect  square  by  adding  a'^x'^, 
but  if  we  add  a'-x'^  we  must  subtract  it  in  order  not  to 
change  the  value  of  the  expression.    We  may  then  write, 

x^  -\-a''x'-  -ha^=x*  +  2a''x'-  -ha^-a^x'^, 
or,  using  a  parenthesis,    =(x* -i-2a'^x'^  -{-a'^)—a'^x'^ . 

Now  it  is  easy  to  see  that  we  have  the  difference  of  two 
squares,  which  we  have  already  learned  how  to  factor. 
Thus  we  have 

X*  -^a\x'-  +a*  =  (x^  +2a'^x'^  -ha*')-^a'^x^ , 
=  (ix'-+a^y-a'^x\ 
=  (x'^  -\-a^-  +ax)(x'^  +a'^  -ax-). 

Examples. 

Find  the  factors  of  each  of  the  following  expressions  : 

1.  x^-\-dx''+Sl. 

To  make  a  perfect  square  of  this  expression  we  must  add  9^^*,  and 
if  we  add  9.v*  we  must  of  course  subtract  9x^  afterward.  Hence  we  get 

We  now  have  the  difference  of  two  squares,  which,  as  we  have 
learned  before,  is  equal  to  the  product  of  the  sum  multiplied  by  the 
difference  of  the  square  roots  of  the  two  squares.   Therefore,  we  have 

=  (.'c'+9-f-3x)(x8-|-9-3.r), 
or,  as  is  perhaps  a  more  natural  arrangement  of  the  terms  of  the  two 
factors,  (;r2-i-3-v-[-9){.r3-:U-f-9). 

2.  x*  +  ^x^-  +  16.  5.   l  +  a'^+a*. 

3.  x^  +  4xy^-\-16y.  6.  x^+x'^-i-1, 

4.  a*x*+4a^x^-j/^-hl6jy\      7.   Ux* -j- 4u'^ v^-'' +  ziH^K 


200  HARDER    FACTORS,   MULTIPLES,   ETC. 

8.  16jt-4-hl6.r22'4  +  16^8        14.  Slx^-{-SQx^y^  +  lC\v^, 

9.  8Li:4+36.r2-j-16.  15.  x^''-^4x'y'^-}-l(yj'\ 

10.  16jt:*+36jt:>2  +  8iy.  16.  Wx'^ +4x^  +  1. 

11.  256  +  144  +  81.  17.  x^y-hx-y^-l-y^. 

12.  81  +  9  +  1.  18.  jt-y+x^j/fi+jK^. 

13.  81  +  9«2-j-«4^  .  ig.  x4jj/%-i+4.r2^''^2  +  16. 

20.  :^  +  .-4-  +  81.  21.  :^  +  9.r2  +  81«^. 

184.  Sometimes  an  expression  assumes  the  form  treated 
in  this  exercise,  after  a  factor  is  removed  from  each  term, 
as  in  the  following  examples  : 

22.  lOmx^  +  lOa^'mx^  +  lOa^m. 

23.  ax^-j-a'^x^-^-a^x. 
a^x^  ,  a^x"-  ,     « 

24.  — — -  +  — --  +  ^6^ 

25.  10jry  +  10.;i:2j/2  +  10. 

26.  _^-+--_^_H-2^8. 

27.  48  +  12^2^2+3^4^4^ 

EXERCISE  92.* 

Expressions  of  the  Form  a^^—b'K 

185.  The  two  general  expressions  a"—b"  and  a"+^", 
where  n  stands  for  any  positive  whole  number,  are  of  so 
much  importance  that,  although  somewhat  difficult  to 
factor  because  of  the  fact  that  n  is  such  a  general  symbol, 
still  we  must  try  to  discover  under  what  circumstances 
expressions  have  factors  of  the  form  a-^b  or  a—b.  We 
begin  with  the  first  of  these  two  expressions,  viz. :  a"—b". 


*  This  exercise  and  the  next  one  are  more  difficult  than  most  portions  of  the 
book,  and  may,  at  the  discretion  of  the  teacher,  be  taken  now  or  postponed  till 
some  later  time  in  the  course. 


EXPRESSIONS    OF    THE    FORM    a'^—b''.  20I 

1.  Divide  a^  —  b^  by  a—b  and  carefully  note  the  form 
of  the  quotient. 

2.  Divide  a^  —  b^  by  a  —  b  and  carefully  note  the  form 
of  the  quotient. 

3.  Divide  a^  —  b^  by  a  —  b  and  carefully  note  the  form 
of  the  quotient. 

4.  Divide  a^  —  b^  by  a—b  and  carefully  note  the  form 
of  the  quotient. 

5.  Without  trj^ing,  would  you  think  that  a'^  —  b^  could 
be  divided  hy  a—b} 

6.  Could  you  guess  the  form  of  the  quotient  of 
(a'-b')^{a-b)} 

7.  Go  through  the  work  of  division  of  (a'^  —  b'^')  —  {a  —  b) 
and  see  if  you  have  guessed  the  right  form  of  the  quotient. 

186.  Now  you  would  probably  guess  that  a"—b" 
could  be  divided  hy  a  —  b  and  that  the  quotient  would  be 

«"-i_(_^"-2^_l_^"-3^2_^^«-4^3  .pother  terms, 
where  the  exponent  of  the  power  of  ^  continually  dimi- 
nishes and  that  of  the  power  of  b  continually  increases 
by  1  in  each  succeeding  term  until  you  arrive  at  b"~'^ . 
A  result  of  this  kind  where  a  series  of  terms  are  written 
in  regular  order  but  where  the  number  of  terms  is  not 
definitely  known  is  sometimes  written  like  this  : 

a"-''  -\-a"-H-i-a"-H^-  -ha'-H^  -\-  .   .   .  +b"-K 

The  dots  (read  afid  so  on)  are  called  the  Sign  of  Con- 
tinuation. They  serve  to  show  that  the  intervening 
terras  are  written  one  after  another  according  to  the  same 
law  as  the  terms  already  written. 

8.  What  would  be  the  term  just  after  a"-'^b^  ? 

9.  What  ^vould  be  the  term  just  before  b"'^  ? 


202  HARDER    FACTORS,   MULTIPLES,   ETC. 

187.  If  the  student  will  carefully  get  fixed  in  mind 
the  meaning  of  these  dots,  we  can  doubtless  go  through 
the  work  of  multiplying  and  see  if  this  expression  mul- 
tiplied hy  a—b  will  produce  a''—b";  the7i,  if  this  should 
be  the  case,  what  was  before  a  guess  becomes  a  certainty. 
For  the  student  will  remember  that  the  expression  we 
are  here  talking  about,  viz. : 

was  a  giiess  at  the  quotient  of  dividing  a"—b*'-  by  a~b, 
and  therefore  if  this  quotient  multiplied  by  the  divisor 
equals  a"—b"  the  guess  was  right.     Now  let  us  multiply 

a—b 

a"+a"-H+a"-''b''^a"-^b-^-{-   ....    ■^ab"-'^ 
-a"-H-a"-H~-a''-H'^-a"-H''-   .   .   .   -b" 


Let  us  stop  a  moment  before  adding  up  these  partial 
products  to  notice  that  there  is  a  sign  of  continuation  in 
each  partial  product. 

10.  In  the  first  partial  product  what  is  the  term  just 
after  a''-'H->  ? 

11.  What  is  the  term  just  before  ab"~^  ? 

12.  In  the  second  partial  product  what  is  the  term  just 
after  — a"-^^^^  ? 

13.  What  is  the  term  just  before  —b"} 

Now  adding  the  above  partial  products  w^e  get  the 
product,  which  is  seen  to  be  a"—b".     Therefore, 

a-^b"=^(^a-bXa"-^+a''--b-^a"-'H''-\-  .   .   .  +b"-^). 

Therefore,  a—b  is  a  factor  oi  a"—b",  whatever  positive 
whole  number  is  represented  by  ?i. 


EXPRESSIONS    OF    THE    FORM    a'' — b"".  203 

* 
187.   As  the  sign  of  continuation  usually  offers  con- 
siderable difficulty  to  the  student,  let  us  study  a  little 
further  the  expression  a:'~^ -\-a''~'^b-\-a"~^b''- -\-  .   .  4-^""^ 
in  which  this  sign  first  appeared. 

14.  Is  «"-i-fa"-2/^+«"-3^'+  .  .  .  -f/^"-i  a  factor  of 
a'^-b"-^.      Why? 

15.  Is  a"—b"  2.  multiple  oi  a''-""  ^-a"-'b-\-a"-^b'^ -\-  .  . 
.   .  +^"-1? 

16.  By  what  expression  will  we  have  to  multiply 
«"-i+«"-2^+a"-3^--f  .   .  .  +b"-'^  to  produce  ^"—<^"? 

17.  Can  a"—b"  be  divided  by  a"~^-{-a"~''b-\-a"~^b^-\-  .  . 
.   .  -\-b"-'? 

18.  What  is  the  quotient  in  example  17  ? 

19.  Will  any  factor  of  a"-i +a"-2^+a"-3^2^  .  .  +b"-^ 
be  a  factor  of  a" — b"  also  ?     Why  ? 

20.  In  a"-'^-\-a"--b-i-a"-^b'^-{-  .  .  .  -j-b"-'^  how  many 
terms  contain  b  ? 

Write  down  5  or  6  terms  and  then  begin  at  the  second  term  and 
count  the  successive  terms,  looking  at  the  exponent  of  /^  while  count- 
ing, and  you  can  doubtless  answer  this  question. 

21.  In  a"-'^+a"-'^b+a'*-^b^-i-  •  .  •  -f  ^"~^  how  many 
terms  are  there  altogether  ? 

Notice  that  each  term  except  the  first  contains  />  and  you  can  prob- 
ably answer  this  question. 

22.  The  first  two  terms  contain  the  common  factor 
a"~'^.     Can  their  sum  be  written  a"~'^{a-\-b)  ? 

23.  Do  the  third  and  fourth  terms  have  a  common 
factor?  What  is  their  H.  C.  F.?  How,  then,  can  their 
sum  be  written  ? 

24.  Do  the  fifth  and  sixth  terms  have  a  common  iactor  ? 
What  is  their  H.  C.  F.?  How,  then,  can  their  sum  be 
written  ? 


204  HARDER    FACTORS,   MULTIPLES,   ETC. 

188.  If  n=b,  the  expression  we  are  considering  is 
a^-\-a^b-\-a'^b'^ -\-ab^ -^b^,  and  if  the  terms  are  grouped 
together  in  sets  of  two  as  far  as  possible,  we  may  write 
(a*+<2^^)-f(^2^2_|_^^3-)_j_^4^  where  there  is  one  term  left 

over  at  the  end  which  cannot  be  put  in  any  group  because 
there  is  no  term  to  go  with  it.  But  if  ?i  —  Q,  the  expres- 
sion we  are  considering  \s  a^  -\- a"^  b -{- a'^  b' -\- a'^  b^ -\-  ab"^  -f-  b^' , 
and  if  the  terms  of  this  are  grouped  in  sets  of  two  as  far 
as  possible,  we  may  write  {a^ -{-a'^b) -\- (^a^b"^ -\- a-b^) 
+  (ab'^-\-b^),  where  there  is  ?io  term  left  over  at  the  end. 

25.  If  in  a"-'^-j-a"-H  +  a''-^b'^-\-  .  .  .  +^"-1  the  terms 
be  grouped  together  in  sets  of  two  as  far  as  possible,  when 
will  there  be  one  term  left  over  at  the  end  and  when  will 
all  the  terms  be  thus  grouped  ? 

Ans7ver:  All  the  terms  can  be  thus  grouped  when  there  is  an  even 
number  of  terms,  i.  e.  when  n  stands  for  an  even  number,  and  one 
term  will  be  left  over  when  the  number  of  terms  is  odd,  i.  e.  when  ;/ 
stands  for  an  odd  number. 

26.  Suppose  n  an  even  number,  group  all  the  terms  of 
^,,_i_j_^„_2^_l_^.-3^2_}_  ^    _    _  -f  ^"-1  in  sets  of  two,  take 

out  the  H.  C.  F.  from  each  group,  and  then  tell  what  is 
a  factor  of  the  whole  expression. 

Notice  that  the  factor  which  you  have  just  found  is  a  factor  of 
rt"-i-|-rt"-2(^-|-(7«-3^3_j_  _|_^«-i  ^w/v  when  n  is  an  even  number. 

27 .  \sa-\-b2i.  factor  of  «" — b"  when  «  is  an  even  number  ? 

See  question  19. 

189.  Now  the  student  can  probably  understand  the 
following  demonstration  : 

a—b 
Hence,  «"-^"=(«-^)(^"-i+^"-2^+^"-^^2+  .  .  +/^"-'). 
The  number  of  terms  of  this  expression,  a''~^ -\- a''~'^ b 
-fa^-s^a.^  .  .  .  +^"-1  is  n,  for  there  are  n—\  terms  that 


EXPRESSIONS    OF    THE    FORM    rt" — ^^  205 

contain  b  (as  can  be  seen  by  beginning  at  the  second  term 
and  counting,  noticing  the  exponents  of  Awhile  counting,) 
and  one  term  that  does  not  contain  b,  so  there  are  ii  terms 
in  all.  Therefore,  when  n  stands  for  an  even  number 
the  terms  of  a''-^ -\-a''-''-b-\-a''-^b'^ -\-  .  .  .  -^b"-^  can  be 
grouped  in  sets  of  two,  and  in  each  set  we  can  take  out 
the  H.  C.  F.  and  express  the  result  as  follows  : 

a:'-''-{a-\-b)^-a"-''b''-{a^b)-\-a"-'^b\a^b)-]r  •  -\-b"--{a-^b) 
Evidently  <^-f^  is  a  factor  of  this  expression,  i.  e.  a  +  b 

is  a  factor  of  «"~^-f«"~^^  +  «"~^t(^-4-  .   .  .  +^"~^. 

But  any  factor  of  «"-i+a"-2/^+a"-'''<^2^  .   .   .  4-^"~Ms 

also  a  factor  of  a"—b"  because  a"—b"  is  a  multiple  oi 

Therefore  ^  +  <^  is  a  factor  of  a"  +  b"  when  71  stands  for 
any  eve7i  number. 

Notice,  this  demonstration  will  not  hold  if  ;^  is  an  odd 
number  because  the  terms  of  a"~'^-\-a"~'^b-\-a"~^b'^-\-  .  . 
.  .  +b"~^  cannot  then  a/l  be  grouped  in  sets  of  two,  for 
one  term  will  remain  over. 

190.  We  have  so  far  reached  the  following  results : 
a"—b"  can  a/ways  be  divided  by  a—b,  and  a"—b"  can  be 
divided  hy  a  +  b  when  71  is  aTiy  eve7i  7iu7nber,  or  stated  in 
another  way,  a—b  is  always  a  factor  of  a"—b"  and  a-\-b 
is  a  factor  of  a"—b"  whe7i  n  is  a7iy  eve7i  Tiumber. 

191.  Thus  it  appears  that  the  difference  between  like 
powers  of  two  numbers  can  always  be  factored,  but  it 
must  not  be  supposed  from  what  has  been  said  that  the 
easiest  and  best  way  to  factor  d"  —  b"  is  always  to  take  out 
the  factor  a—b  first,  for  it  may  be  that  the  remaining  ex- 
pression after  this  factor  has  been  removed  will  be  harder 
to  factor  than  the  one  we  started  with  would  be,  if  we 
proceed  to  take  out  some  other  factor  first.  This  will  be 
fully  seen  in  the  examples  which  follow. 


206  HARDER    FACTORS,   MULTIPLES,   ETC. 

Examples. 

Factor  as  far  as  j^ou  can  each  of  the  following  expres- 
sions : 

I.  a^-b"^. 

This  we  know  has  a  factor  a—b,  and  also  because  the  exponent  is 
even  a  factor  a-\-b,  and  if  we  take  out  each  of  these  factors  in  turn 
we  would  have  left  a^-^b^. 

Therefore,  a^~b^  =  [a-b){a-{-h){a^-[-b^). 

Or  we  might  proceed  thus:  a'^—b^  may  be  regarded  as  the  differ- 
ence of  two  squares  and  factor  accordingly. 

Therefore,     a^-b^={a^Y -(b'^Y  =  [a'^  —  b^){fl^-\-b''^) 
z=z\a-b){a-^b){a^+b^), 
the  same  as  before,  as  it  ought  to  be. 


2, 

x^^-l. 

6. 

16^'*— 81. 

10, 

i6x^-ie,j^\ 

3. 

x^-y. 

7. 

l-jr*. 

II. 

.^*J/*— ?/*Z'*. 

4. 

X^J^-2\ 

8. 

l-x'^yK 

12. 

5. 

r^s^-t\ 

9. 

16a^-x^y\ 

13. 

u""      x'^ 

14.   a^  —  b^. 
From  the  general  discussion  which  precedes  we  know  that  the  fac- 
tors of  this  are  a—b  and  a^-^a^ bA-a~ b'^ -\-ab'^ -\-b^ ,  and  this  is  the  only 
way  we  have  at  present  to  factor  this  expression. 

15.  x^'  —  \.  17.  x^y'^—z'"".  19.   \—x^\ 

16.  ^5_^5  18.  2/5z/5  — 32.  20.  243—32. 

21.   Zlx^y^  —  ii^v^w^.      23.   a''::f^y^  —  ?)'^lu^v^w^. 

11.   — < ~.  24.   -^. ~. 

z^         w^  y^  y^ 

25.   a^  —  b^. 
This    may   be  regarded  as  the  difference  of  the  two  cubes,    and 
hence  we  may  write 


EXPRESSIONS    OF    THE    FORM    rt" — <^«.  20/ 

Each  of  these  factors  is  a  form  already  treated,  and  so  we  know  the 
factors  of  each  factor,  viz: 

a^  —  (,i  —  {a-h){a  +  l>) 
and  a*  -^a^h^  ■^h^  =  {a^  +b^  +ab){n^ +h^  -ab). 

Therefore,  a^  ^  h^  =(a-b){n+b){a^  +b^  -^ab){a^  +/>^  -ab). 

Or,   if   we  prefer,   we  may  regard    a^—b^  as  the  difference  of   two 
squares,  and  hence  may  write  : 

a^—b^={a^)^-{b^Y-{a^-b^){a^  +  b-^). 
Here  again  each  factor  is  a  form  already  treated,  and  so  we  know 
the  factors  of*each  factor,  viz: 

a^-b^  =  {n—b){a^  +  ab  +  b^) 
and  ,r^  +  b^=:{a  +  b){a^-ob  +  b*). 

Therefore,  a'^-b^  =  [a  —  b)(,a-irb)[a^-i-ab  +  b'^){a*—ab  +  b^). 

This  agrees  with  the  result  obtained  before,  as  it  ought  to  do. 

26.  x^  —  l.  28,   — g— w'^.r^.         30.  x^y^2^  —  \. 


27.  x^y^—2^.  29.   7i^x^—r^y^'2'^  31 


X^         |,  6 


32.   a'^  —  b'^. 

From  the  general  discussion  which  preceeds  we  know  that  the  fac- 
tors of  this  ^rea  —  b  and  a^  +a^b  ^ro^b-  -\-o^b-^  +a-b*  \-ab^  +b^,  and 
these  are  the  only  factors  of  a"'  —b'  that  we  can  find  now 

33-  x'^y"'—!.  35.  x'^y'^—z\ 

x"^        11^  ^      ^         ?/77;7 

34.    -y--?-  36.   1 


yi  1}i  -  W'^X'^' 


37.  a^  —  b^. 

This  may  be  considered  either  as  the  difference  of  two  fourth 
powers,  or  the  difference  of  two  squares.  Taking  it  in  the  latter  way 
we  may  write, 

a^  —  b^—(a^-b^){a'^A.b^^. 
The  first  of  these  factors  has  been  considered  before,  so  we  know 
how  to  factor  it.     Therefore  we  have, 

a^-b^-{a^-b^){a^Jrb^) 

={a-b){a^-b){a'^^.b^'){a*'  +-<5*). 


208  HARDER    FACTORS,    MULTIPLES,   ETC. 


38. 

x^y^  —  1. 

40.    X^—7C^V^. 

39. 

x^      u^ 

y%               ^8* 

4^-     .<-     .s- 

42. 

«9- 

-dK 

This  may  be  regarded  as  the  difference  of  two  cubes,  viz; 

(a3)3_(<^3)3. 

Therefore,  a^ -d^={a^—d^){a^ +a^d^ +  1?^) 


43.  x^y^-z\ 

45.   u^v^-sH\ 

x^      . 

44-        y,             1. 

71^       ^9 

EXERCISE  93. 

Expressions  of  the  Form  a^'-\-b'K 

192.  In  the  previous  exercise  we  found  that  a''—b"' 
was  always  divisible  hy  a—b.  Now  this  dividend,  d"—b'\ 
means  that  the  71  th  power  of  b  is  to  be  subtracted  irom 
the  7ti\i  power  of  a,  and  the  divisor,  a—b,  means  that 
the  number  represented  by  b  is  to  be  subtracted  from  the 
number  represented  by  «,  and  in  both  dividend  and 
divisor  a  and  b  may  stand  for  either  positive  or  negative 
numbers.  We  may  then  write  a7iy  numbers  or  letters  we 
like  in  place  of  either  a  or  ^  or  both. 

Suppose,  then,  we  take  the  case  where  n  stands  for  an 
odd  number  and  write  —b  in  place  of  b  in  both  dividend 
and  divisor.  Now,  because  71  is  an  odd  number, 
(^—by=  —  b'\  and  if  this  expression,  —b",  be  subtracted 
from  a"  we  get  a''-\-b*'  for  a  dividend,  and  if  —  ^  be  sub- 
tracted from  a  we  get  «  f  ^  for  a  divisor. 

Therefore,  a-\-b  is  a  divisor,  i.  e.y  o.  factor ,  of  a"-\-b" 
when  n  is  a7iy  odd  7iU7nber, 


EXPRESSIONS    OF    THE    FORM    «"  +  ^«.  209 

193.  To  find  out  whether  a-\-bova—dis  a  factor  of 
a"-{-d"  when  71  is  eve?i,  a  little  preliminary  stud}^  is 
necessary. 

1.  Is  5  a  factor  of  55  ?  Is  5  a  factor  of  15  ?  Is  5  a 
factor  of  55  +  15  ?     Is  5  a  factor  of  55—15  ? 

2.  Is  a  a  factor  of  ax?  Is  a  a.  factor  of  ajy}  Is  a  a 
factor  of  ax-\-ay  ?     Is  a  a.  factor  of  ax— ay  ? 

3.  If  a  number  is  a  factor  of  each  of  two  other  num- 
bers, is  it  a  factor  of  the  sum  of  those  numbers  ?  Is  it  a 
factor  of  the  difference  of  those  numbers  ? 

4.  If  one  number  is  a  factor  of  a  second  and  not  of  a 
third,  can  the  first  number  be  a  factor  of  the  sum  of  the 
second  and  third  numbers  ?  Can  the  first  number  be  a 
factor  of  the  difference  of  the  second  and  third  numbers  ? 

5.  When  n  is  even  is  a-\-l?  a.  factor  of  a"—b"?  When  n 
is  even  is  a-\-d  a.  factor  of  2d"  ?  When  ?i  is  even  is  a-\-d 
a  factor  of  (a"-d")-i-2b"? 

6.  Since  (ia"—b")  +  2b"=a"-\-d"isa  +  bsL{2iCtovofa"-\-b" 
when  n  is  even  ? 

7.  Is  a—d  a  factor  of  a"—d"  ?  Is  a—d  a  factor  of  2d"  ? 
Is  a-d  a  factor  of  (^a"-d"-)-\-2b"  ? 

8.  Since  {a"—d")  +  2d"=a"  +  d"  is  a—b  a  factor  oi 
a"-\-d"J 

194.  Questions  6  and  8  if  rightly  answered  and  un- 
derstood, give  us  the  fact,  that  a" +  5"  is  never  divisible  by 
a—d  and  a"-\-d"  is  not  divisible  hy  a  +  d  when  n  is  even. 

196.  We  previously  found  that  a"-\-b"  is  divisible  by 
a-\-d  when  ?i  is  an  odd  number. 

14 


2IO  HARDER    FACTORS,   MULTIPLES,   ETC. 

196.  What  has  been  found  in  the  previous  exercise 
and  in  this  one,  may  now  be  stated  as  follows: 

a"—b"  is  always  divisible  by  a—b. 

a" — b"  is  divisible  by  a-\-b  whefi  n  is  an  even  number^ 
but  not  when  n  is  an  odd  number. 

a"-\-b"  is  divisible  by  a-{-b  ivhen  n  is  an  odd  number,  bia 
not  when  n  is  an  even  7iumber. 

a"-\-b"  is  never  divisible  by  a — b. 

197.  Although   a''-^b"  cannot  be  divided   by  either 

a-\-b  or  a—b  when  n  is  even,  yet  it  is  not  stated  that  in 
this  case  a''-\-b"  has  710  factors,  for  sometimes  other  factors 
can  be  found,  as  will  be  seen  by  the  examples  to  follow 
and  the  explanations  accompanying  them. 

Factor  each  of  the  following  expressions  : 

9.  a^-\-b^. 
By  the  general  discussion  we  know  that  this  has  a  factor  (X-\-b,  and 
another  factor,  found  by  division  of  a^-\-b^  by  a-\-b,  is 

10.  x^-\-\.  12.  a'^^x'^-^-b'^'y^.     14.  n^x^ -^^^x^y'^'z^ , 

11.  x'^ y''' -\- z^ .      13.   l  +  32;»;^.  15.   a^ -\-^lic^ v''' w^ . 

^    ;r^  .  32  „    a^x'"  ,  b'^z^ 

16.  -5-  +  -^.  18.   — 5--  +  --^. 

yO  ^O  yb  yb 

a^     b^  .   ,  a^ 

17,  ^~,  +  -,.  19.   1  +  32. 

20.   a^-\-b^. 
This  may  be  regarded  aS  the  sum  of  two  cubes,  and  therefore  we 
may  write      a^  ^b^-{a^Y  +{b^Y-{a^  ^b^){a^-a^b'^  ^b^). 

21.  x^-\-y^.        23.  «6^^  +  64^^j^.  25.   64:-\-x^y^z^. 

22.  x^-\-l.  24.  «^z^^4-G4.  26.  x^y'^z^-^x^y^w^. 

27.  ^  +  1.  28.   -'4-4- 

'       y6  ^6    '   ^6 


EXPRESSIONS    OF    THE    FORM    a^'  +  d".  211 

29.   a'^-hd'^. 
By  the  general  discussion  we  know  that  this  has  a  factor  a-\-d,  and 
the  other  factor  obtained  by  division  will  be  found  to  be 

30.  x'^-j-y.  32.  71'^v'^+x'^y^.      34.   l-{-x''yz'^. 

31.  u-'  +  l.  33.  aH'^c'^ -i-128.     35.   128  +  1. 

36.    -   -J ~.  37.    -7  +  —-. 

v^     y^  y^     x^ 

38.   a^  +  b\ 
This  may  be  regarded  as  the  sum  of  two  cubes,  and  hence  we  may 

write  rt9+(^»=(«3)3+(/;3)3_(^3_,.^3)(rt6—^/3/'3 +/;«), 

=  (^a  +  l>){a^-al>-\-l>''){a^—a^b'^+i)^). 

39.  x'^  +  l.  41.  21^+vHv^xK     43.  ^  +  -9. 

40.  x^y^+z^,        42.  a9;i:9+<^9j/9^9.44.  ^4-^. 

45.      a^«  +  ^^«. 

This  may  be  regarded  as  the  sum  of  two  fifth  powers,  and  hence 
we  may  write, 

46.  x''-{-y^\        48.   1024  +  .r^0j/io.5o.  |J-^4-^o- 

47.  a^'+l.  49.   1024  +  1.  51.  ^5  +  1. 

198.  In  the  expressions  considered  in  this  exercise 
and  the  preceding,  the  exponent  in  each  term  is  the  same, 
but  the  methods  used  enable  us  in  some  cases  to  find  fac- 
tors of  an  expression  in  which  the  exponents  are  not  the 
same  in  each  term,  as  for  instance  in  the  expression 
^5_|_^io  '^\^^  exponents  are  5  and  10  respectively,  but 
we  may  consider  b'^  ^  =^ (^b'^^^  and,  therefore,  a^  +  b^^  may 


212  HARDER    FACTORS,   MULTIPLES,   ETC. 

be  considered  the  sum  of  two  fifth  powers,  viz:  a^-\-  {b'^')^ , 
and  may  be  factored  as  in  example  9,  using  b"^  the  same 
as  b  was  used  before. 

We  add  a  few  miscellaneous  examples  on  the  last  two 
exercises,  where  in  some  cases  a  factor  must  be  removed 
from  each  term  before  the  expression  is  in  the  form 
treated  in  either  of  these  last  two  exercises. 

52.  3^4-48.  61.  x^-a^y^"^. 

53.  2a^  —  Z2a''x^.  62.   r^x^'^  —  bx^y^'^. 

54.  3;i:*-3yi^  63.    lOa^-\Qb^c^. 

55.  x^-\-a^y^\  64.    {x+yY—i^x—yy. 

56.  ^u'^v^-Zu^x^\  65.   Z{x^ +y^y -Z{x^ -y^y 

4  4  ^^     {x^-y^y  64 


{x'^-^y^y     {x^'-y'^y       '  64  (^3_y^6- 

58.  {a  +  by-{a''-b''y.       67.   «i8-ai2. 

59.  («2_^^2-)4_(^2_^2)4_  68.   3;«.;ci4_3w8j/7^ 

32  32       •        ^  «6- 

EXERCISE  94. 

Miscellaneous  Factors. 

199.  Some  expressions  are  quite  difficult  to  factor  un- 
less one  sees  how  the  expression  may  be  changed  in  form 
by  rearranging  or  grouping  of  the  terms,  or  by  adding 
and  then  subtracting  the  same  expression,  or  by  some 
other  device  to  change  the  expression  into  a  form  that 
will  be  recognized  as  coming  under  some  case  already 
treated.  To  help  the  student  see  some  of  the  devices, 
we  take  some  of  these  irregular  expressions  and  work 


MISCELLANEOUS    FACTORS.  21  3 

them  out.  The  student  is  advised  to  go  through  the  ex- 
planation given,  to  see  just  what  is  done,  and  if  possible, 
ivhy  it  is  done,  and  after  two  or  three  cases  to  cover  up 
the  explanation  given  and  see  if  some  method  suggests 
itself;  if  not,  see  how  the  work  is  started  and  then  try  to 
complete  it  without  looking  at  the  rest  of  the  explana- 
tion. If  still  unable  to  do  the  work,  study  the  whole 
explanation  given. 

1.  Let  us  factor  a3  +  <53-f  ^3  ^3<^2^+ 33^2^ 
We  may  arrange  the  terms  thus  : 

where  the  last  four  terms  form  a  perfect  cube,  viz:  the 
cube  of  b-\-c.  Therefore,  the  given  expression  may  be 
written  a^ -\-{b-\-cY . 

We  now  have  the  sum  of  two  cubes,  and  therefore  the 
factors  are  ^  +  ^-f-^and  a'^  —  a{b-\-c)-\-{b-\-cYi  ox  a-\-b-\-c, 
and  a'^^-b'^-\-c^—ab—ac-\-1bc. 

Hence,  a^^-b""  ^-c^  -^Zb'^c-VUc'^ 

=  (a-\-b-i-cXa''-^b^-+c'--ab-ac+2bc), 

2.  l^t  us  (sictor  a'^  +  b^+c'^+3a'^b-\-Sab^. 
We  may  arrange  the  terms  thus : 

a^-\-SaH  +  Sab''+b^-hc^, 
where  the  first  four  terms  form  a  perfect  cube,  viz:  the 
cube  of  a  +  b.     Therefore,  the  given  expression  may  be 
written  (a-^b)^+c^. 

We  now  have  the  sum  of  two  cubes,  and  therefore,  the 
factors  are  a-\-b-\-c  and  (a-\-b)^^{a-^b)c-\-c^ ,  or  a+b-\-c, 
and  a"^  -^b^  -{-c"^  -j-2ab—ac—bc. 

Hence,  a^-hb^+c^+SaH+Sab^ 

^{a-\-b-{-c)(a'^  ^b"^  ^c"^  +2ab-ac-bc). 


214  HARDER    FACTORS,   MULTIPLES,   ETC. 

3.  Let  US  factor  the  expression  a'^-\-2a^d—a'^'-2a5^ 
This  can  be  arranged  thus  : 

where  it  is  plain  that  a'^^b'^  is  a  factor  of  each  of  the 
three  parts  into  which  the  expression  is  grouped. 
Taking  out  this  factor,  the  expression  may  be  written 

or  (^2_^2)|-(^2  4.,2^^_|.^2)_i]^ 

or  {a''-b''-)\{a-Vby-\\ 

Each    of   these    factors  is  itself  the    difference  of   two 

squares,  and  hence  each  may  be  further  factored  thus  : 

Therefore  we  may  write, 

a^-^la'^b-a'^-lab^-^b'^-b^ 
=  {a-b^{a^-b){a^b-V){a-Vb-\-V), 

4.  Let  us  factor  the  expression  a2^2_^2^2_^^2_^2^ 
This  may  be  written  thus  : 

(«2_32)^2.^^2_^2^ 
or  (^?2_^2)^2_^^2_^2)^ 

or  («2__^2)(_^2_1)^ 

or  {a-b){a-^bXx-r){x-\-r)^ 

5.  Let   us   factor   the   expression    a^  —  a"^  -\-  b^  —  h'^ 
—2a'^b''--2ab. 

This  may  be  written  thus  : 

{a^-2aH''-^b^)-{a''-^2ab  +  b-'),    ' 
or  {a'-b''y--{a-\-by. 


MISCELLANEOUS    FACTORS.  21$ 

This  is  the  difference  of  two  squares,  and  hence  may  be 
written  as  the  product  of  the  sum  into  the  difference  of 
the  two  numbers  which  are  squared.     Hence, 

=  [(«2_^2)_(^  +  ^)][-(^2_^2)_^(^_|.^)] 

=  {a+b){a-b-l)(a  +  b){a-b+V), 

6.  I^et  us  factor  a"  +  /^^+<:^— 3«<^<:. 
This  may  be  written  thus  : 

a^  +  {b^  -irWc+'^bc''  +c''')-Wc-Uc''-Zabc, 
or  a^  +  {b-{-cY~Uc{a-\-b+c). 

The  first  two  terms  are  now  the  sum  of  two  cubes,  and 
hence  contain  the  factor  a-\-b+c,  and  this  is  also  a  factor 
of  the  last  term,  as  is  very  evident.  We  may  therefore 
write  the  expression  considered  in  the  form 

ia  +  b+c)[_a''-a(^b+c)^-(^b+cy^-^bc{a  +  b+c), 
or  (a-{-b-^c){a'^—ab—ac-{-b-  —  bc-\-c''-). 

7.  lyet   us   factor    a^-\- b^-\- c^-\- ab'^-\- ac'^-\- a'^b+ a'^c 
^-bc-'-^-b'^c. 

This  may  be  written  thus  : 

In  the  first  group  ^'^  is  a  factor  of  each  term,  in  the 
second  group  ^^  is  a  factor  of  each  term,  and  in  the  third 
group  c^  is  a  factor  of  each  term.     Hence,  we  may  write 

a''{a-Vb^-c)-\-b''{a^-b^c^-\-c''{a^-b^-c). 
And  now  evidently  a-\-b-\-c\s>  a  common  factor  through- 
oat.     Hence,  the  original  expression  equals 


2l6  HARDER    FACTORS,   MULTIPLES,   ETC. 

8.  12;t:2-36.r4-24. 

9.  5;r2+5;i:— 10. 

10.  x'^j/-i-xj}^—x'^—x—2y-\-2. 

11.  (a  +  b  +  cy-(a-d-cy.     14.  x^-^-Sj"^ -\-x-\-2y. 

12.  (c-^dy-^(c-dy.  15.  «5_8^2^3_ 

13.  x''-iy^+x-2j^,  16.  500a'2j/-20j/\ 

17.  1— ^2^-2— /^2^2_j_2^^^_y^ 

18.  a^-j-x''-(^y''-i-2^-')-2(ij^2-ax^. 

19.  5.:^;4- 15.^3 -90;i;2^ 

EXERCISE  95. 

H.  C.  F.  OF  Expressions  which  Can  be  Factored. 

200.  The  highest  common  factor  of  several  expressions 
has  already  been  defined,  (see  Art.  82,)  and  a  method  of 
finding  the  H.  C.  F.  of  several  monomml  expressions  has 
been  given.  We  now  take  up  the  H.  C.  F.  of  poly- 
nomials. 

201.  The  first  method  to  consider  is  exactly  the  same 
as  that  presented  in  the  case  of  monomials,  viz. : 

/Resolve  each  expression  into  its  priyne  factors  ajid  take  the 
product  of  all  those  which  are  common  to  all  the  expressions. 

By  this  method  we  can  readily  find  the  H.  C.  F.  of  any 
expressions  which  can  be  readily  resolved  into  their 
prime  factors. 

Examples. 

Find  the  H.  C.  F.  of  each  of  the  following  sets  oi 
expressions  : 

1.  x"^ •\- xy  2,wA  x"^ —y"^ .  3.  j»;^— j/''^  and  ^2— jj/2. 

2.  2.^2  — 2;rK  and  3.;t2—3>/2     ^    x^ -\-y^  2.viA  x'^—y'^ . 


SECOND    METHOL    OF    FINDING    H.   C.   F.         21/ 

6.  na'^x'^y-Aa^xy''  2inA^0a''x^y'^-10a''x''y^ . 

7.  8«3^V-12^2^^3  and  6«^V+4^<^V2. 

8.  ;r2-2;ir-3and  Jt-2H-ji--12. 

9.  2a{a''-b'')  2.n&Ab{a-by. 

10.  3:i:=^+6.r2-24j»r  and  Gx^-QG;*;. 

11.  j»:2— 9;»;— lOand  ;r2+4ji-+3. 

12.  jt:2— 4  and  ;f2— 5;i:+6.   14.  a'^+x— 6  and  x-— 3a'4-2 

13.  rt!6  — ^6  and^*  — ^4.         15.  -r^+.r+l  andjt-'^  — 1. 

16.  2:t=^+2,     3a'2-3,  and  x''-{-Zx+1. 

17.  jf2— 3^+2,     x-—Q>x  +  S,  and  ;r2+ji-— 6. 

18.  ;i;2j/-2-^^  x''-y'^-dxyz-^22'',2indx^y^-y2^. 

19.  ;t:4-8;»;2  +  16and%r3j/3  4.4_;^2y_|.4^-^3 

20.  ;^;3— ;i:2-f  jtr— 1,  x^ ^x^—x'^—x,  and ^t.^ +2;r— 3. 

EXERCISE  96. 

H.  C.  F.  OF  Expressions  not  easily  Factored. 

202.  The  method  used  in  the  preceding  exercise  for 
finding  the  H.  C.  F.  is  not  appropriate  when  the  expres- 
sions given  are  not  easily  factored.  In  case  the  expressions 
given  are  not  easily  factored,  the  method  to  be  pursued 
depends  upon  a  few  principles  which  must  be  understood 
before  the  method  is  given. 

1.  If  an  expression  be  multiplied  by  some  number  or 
expression,  does  the  product  contain  all  the  iactors  the 
original  expression  contained  ? 

2.  Does  the  product  contain  any  factors  the  original 
expression  did  not  contain  ? 


2l8  HARDER    FACTORS,   MULTIPLES,   ETC. 

3.  Are  the  factors  of  an  expression  also  factors  of  any 
multiple  of  that  expression  ? 

4.  If  one  expression  is  a  factor  of  two  other  expres- 
sions, will  it  be  a  factor  of  the  sum  of  those  expressions? 

5.  Will  it  be  a  factor  of  the  difference  of  those  ex- 
pressions ? 

6.  Will  it  be  a  factor  of  any  multiple  of  the  first  ex- 
pression plus  any  multiple  of  the  second  expression  ? 

7.  Will  it  be  a  factor  of  any  multiple  of  the  first  ex- 
pression minus  any  multiple  of  the  second  expression  ? 

203.  If  these  questions  are  understood,  w^e  may  state 
the  two  principles  to  be  used  in  finding  the  H.  C.  F.  as 
follows  : 

/.  A7iy  factor  of  an  exp?'essio7i  is  a  fad  or  of  any  miiHiple 
of  that  expression. 

II.  A7iy  coimnon  factor  of  two  expressions  is  a  factor  of 
the  sum  or  difference  of  those  expressions  or  of  the  S7im  or 
difference  of  any  multiples  of  those  expressio?is. 

204.  Our  problem  is  to  find,  not  merely  a  co7nmon 
factor,  but  the  highest  common  factor  of  two  or  more  ex- 
pressions. This  problem  can  often  be  considerably  sim- 
plified by  first  taking  out  from  each  of  the  given 
expressions  all  moyiomial  factors  and  finding  the  H.  C.  F. 
of  these  factors,  if  there  be  any,  and  then  find  the  H.  C. 
F.  of  the  remaining  portions  of  the  given  expressions. 
After  this  has  been  done  w^e  must  multiply  the  H.  C.  F. 
of  the  monomial  factors  by  the  H.  C.  F.  of  the  remaining 
polynomial  factors,  when  we  will  have  the  whole  of  the 
H.  C.  F.  of  the  given  expressions. 

I.et  us  find  the  H.  C.  F.  of 

and  3^r3_^;t;2_4^_20.  (2) 


SECOND    METHOD    OF    FINDING   H.   C.   F.         219 

Neither  of  these  expressions  have  atij^  7nonomial  factors, 
therefore  the  H.  C.  F.  of  them  will  have  no  monomial 
factors. 

8.  Will  any  factor  of  expression  (1)  be  a  factor  of  3 
times  this  expression,  i.  e.,  oi 

3x3-3;c2_i2?  (3) 

9.  Will  any  common  factor  of  (1)  and  (2)  be  a  common 
factor  of  (2)  and  (3)  ? 

10.  Will  the  H.  C.  F.  of  (1)  and  (2)  be  the  H.  C.  F. 
of  (2)  and  (3)  ? 

11.  Will  any  common  factor  of  (2)  and  (3)  be  a  factor  of 
their  difference,  which  is 

12.  Will  the  H.  C.  F.  of  (2)  and  (3)  be  the  H.  C.  F. 
of  (3)  and  (4)  ? 

13.  Will  the  H.  C.  F.  of  (1)  and  (2)  be  the  H.  C.  F. 
of  (3)  and  (4)  ? 

Now  remember  that  the  H.  C  F.  of  (1)  and  (2)  can 
have  no  monomial  factors,  and  therefore  the  H.  C.  F.  of 
(3)  and  (4)  can  have  no  monomial  factors,  and  the  next 
question  can  be  answered. 

14.  Will  the  H.  C.  F.  of  (3)  and  (4)  be  the  H.  C.  F.  of 

and  x'^-x-l't  (6) 

15.  Will  the  H.  C.  F.  of  (1)  and  (2)  be  the  H.  C.  F. 
of  (5)  and  (6)  ? 

16.  Will  any  factor  of  (G)  be  a  factor  of  x  times  this 
expression,  which  is 

;j;3_^2_2jr?  (7) 

17.  Will  the  H.  C.  F.  of  (5)  and  (6)  be  the  H.  C.  F. 

of  (5)  and  (7)  ? 


220  HARDER    FACTORS,   MULTIPLES,   ETC. 

i8.  Will  any  common  factor  of  (5)  and  (7)  be  a  factor 
of  their  difference,  which  is 

2x-A  ?  (8) 

19.  Will  the  H.  C.  F.  of  (5)  and  (7)  be  a  factor  of  (8)  ? 

20.  Will  the  H.  C.  F.  of  (5)  and  (7)  be  the  H.  C.  F. 

of  (7)  and  (8)  ? 

21.  Will  the  H.  C.  F.  of  (7)  and  (8)  be  the  H.  C.  F.  of 

x^-x-2  (9) 

and  x—2?  (10) 

22.  Will  the  H.  C.  F.  of  (9)  and  (10)  be  the  H.  C.  F. 
of  (5)  and  (7)  ? 

23.  Will  the  H.  C.  F.  of  (9)  and  (10)  be  the  H.  C.  F. 

of  (5)  and  (6)  ? 

24.  Will  the  H.  C.  F.  of  (9)  and  (10)  be  the  H.  C.  F. 
of  (1)  and  (2)  ? 

25.  What  is  the  H.  C.  F.  of  (9)  and  (10)  ? 

26.  What  is  the  H.  C.  F.  of  (1)  and  (2)  ? 

205.  I^et  us  now  look  over  and  see  how  these  various 
expressions  came  about.  Expression  (1)  was  multiplied 
by  3  to  get  expression  (3).  Where  did  this  multiplier  3 
come  from  ?  We  notice  that  if  we  divide  (2)  by  (1)  the 
quotient  is  3  and  the  remainder  is  4x'^^4x—8,  which  is 
numbered  (4).  This  expression  (4)  contains  a  monomial 
factor  4,  which  when  rejected  leaves  x'^—x—2,  the  ex- 
pression numbered  (6).  The  work  so  far  can  be  arranged 
as  follows  : 

;^3_^2__4  )  3^34.  ^2_4^_20  (  3 
Zx^-Zx''  -12 

4  )  4.^2 -4.y--  8 

x'^—  x—  2  V 


SECOND    METHOD    OF    FINDING    H.   C.   F.         22  1 

We  now  deal  with  (5)  and  (1),  or  what  is  the  same 
thing,  (5)  and  (6).  These  two  expressions  have  the  same 
H.  C.  F.  as  (1)  and  (2)  and  are  easier  to  deal  with  be- 
cause the  highest  exponents  in  these  two  expressions  are 
2  and  3  instead  of  3  and  3  as  before.  Thus  our  problem 
is  a  little  simpler  than  before. 

Expression  (6)  was  multiplied  by  x  to  produce  expres- 
sion (7).  But  where  did  this  multiplier  x  come  from? 
Notice  that  if  we  divide  expression  (o)  by  expression  (6) 
the  quotient  is  x  and  the  remainder  is  "Ix—A:,  which  is 
the  expression  numbered  (8).  This  expression  (8)  con- 
tains the  monomial  factor  2,  which  when  rejected  leaves 
.r— 2,  the  expression  numbered  (10).  This  second  part 
of  the  work  can  be  arranged  as  follows  : 

x'^—x—^  )  x'^—x'^  —4  (  X 

x^—x'^  —  2x 

2y2^^ 
x-'Z 
We  now  deal  with  (9)  and  (10)  instead  of  (5)  and  (6). 
These  two  expressions,  (9)  and  (10),  have  the  same 
H.  C.  F.  as  (5)  and  (6)  and  are  easier  to  deal  with  be- 
cause in  these  the  highest  exponents  are  1  and  2  respec- 
tively instead  of  2  and  3  as  before.  We  can  even  find  the 
H.  C.  F.  of  these  expressions,  (9)  and  (10),  by  the 
method  previously  used,  or  we  can  keep  on  with  the 
process  so  far  employed  and  divide  expression  (10)  by 
expression  (9).  This  third  part  of  the  work  can  be 
arranged  as  follows  : 

x-2  )  x''-  x-2  )  x+1 
x^--2x 

x-2 
x-2 

As  this  division  is  exact,  the  H.  C.  F.  of  (9)  and  (10) 
is  x-2. 


222  HARDER    FACTORS,   MULTIPLES,   ETC. 

206.  Thus  it  appears  that  the  process  of  finding  the 
H.  C.  F.  of  two  expressions,  when  they  cannot  be  readilj^ 
factored,  is  to  divide  one  expression  by  the  other,  the  divisor 
by  the  remai7ider,  the  last  divisor  by  the  last  reniai7ider,  and 
so  on  until  there  is  7io  remainder.  The  Last  divisor  is  the 
H.  C.  F. 

It  must  be  remembered  that  this  process  is  not  to  be 
employed  until  all  the  monomial  factors  are  first  taken 
out  of  the  given  expression,  so  that-  the  H.  C.  F.  of  the 
remaining  portions  of  the  given  expressions  contains  no 
monomial  factors.  This  being  done,  we  may,  at  any 
stage  of  the  process  just  described,  remove  from  any 
dividend  or  divisor  any  monomial  factor  we  please  ;  and 
it  will  usually  be  best  to  remove  them  whenever  we  can. 

All  the  work  of  the  preceding  example  may  be  arranged 
as  follov/s : 

x^-x'^—^  )  3;^^+  x2_4^__20  (  3 
^x^-Zx''  -12 

4  )  4;^2_4-^;_  8 

xi-  x—  2 

x'^—x—^  )  x^—x'^  —4  (  X 

x^—x'^—2x 

2  )  2.r-4 

x-2 

x-2  )  x^-  x-2  (  x+l 
x^-—2x 

x-2 

Ivet  us  find  the  H.  C.  F.  of 

4.x^-^x^-2ix-9 
and  Sx^-2x^-5Sx-S9. 


SECOND    METHOD    OF    FINDING    H.   C.   F.         223 

As  there  are  no  monomial  factors,  we  may  arrange  the 
work  thus  : 

4x''-Sx^-2ix-9  )  Sx^-2x''-5Sx-S9  (  2 
Sx^-(3x''-4Sx-18 

4:x'^-5x-21  )  4^3_3^2_24;ir-9  (  x 

2;»;2-  3;»;-9 

2;ir2-3;r-9  )  4x'^-bx-21  (  2 
4.r2^6;tr-18 
^-  3 

x-^  )  2:r2-3;r-9  (  2;t:+3 
2x^—^x 

Zx—9 
Zx-9 


207.  It  will  sometimes  happen  that  the  numerical 
coefficients  are  such  that  the  first  term  of  the  expression 
used  for  a  dividend  is  not  exactly  divisible  by  the  first 
term  of  the  divisor,  in  which  case  we  multiply  the  divi- 
dend by  the  smallest  number  that  will  make  the  division 
exact.  This  peculiarity  is  illustrated  by  the  following 
example  : 

Find  the  H.  C.  F.  of 

3^3-4^-4-3^-2 
and  2a^  —  Za^-{-a''--\-a—l. 

Here  we  use  the  first  expression  for  a  divisor  and  the 
second  for  a  dividend,  and  evidently  the  first  term,  2a^, 
of  the  dividend  is  not  exactly  divisible  by  Za^,  the  first 
term  of  the  divisor,  so  we  multiply  the  dividend  by  3; 
and  the  work  may  be  arranged  thus  : 


224  HARDER    FACTORS,    MULTIPLES,   ETC. 

3^ 

3a3_4a2+3a-2)6«4_9^3_|_3^2_^3^_3  ^2a 

—  a^ — 8<^^  +  7« — 3 

In  a  case  like  this,  where  the  first  term  of  the  remain- 
der has  a  minus  sign,  we  change  all  the  signs  before 
using  it  as  a  divisor.  TRis  is  equivalent  to  multiplying 
by  —1.     Making  the  change,  we  continue  as  follows  : 

a^-\-Sa''-7a  +  S^Sa^-  4a'' -h   Sa—  2(3 
3^^+  9^^-21^+  9 
-lSa''+24:a-n 

Here  again  the  first  term  of  the  remainder  having  a 
minus  sign,  we  change  all  the  signs  of  the  remainder 
before  using  it  as  a  divisor  ;  but  even  then,  the  first  term 
of  the  expression  we  are  to  use  as  a  divisor  not  being 
divisible  by  ISa'^,  we  multiply  the  dividend  (which  was 
the  divisor  in  the  operation  just  performed)  by  13,  so  that 
the  division  will  be  exact,  and  continue  as  follows  : 

a^-j-   3«2_     7a-\-  3 

13 

13a2_24«-fll  )  13^34-39^2_  91^-1.39  (  a+i 
13^3-24^2+   11^ 

63a  2 -102^4-39 
52a''-  96^  +  44 
11^2-     6a-  5 

Again,  when  we  use  this  remainder  for  divisor  and  this 
divisor  for  dividend,  the  first  term  of  the  new  dividend 
is  not  exactly  divisible  by  the  first  term  of  the  new 
divisor,  so  we  multiply  again  by  such  a  number  that  the 
division  will  be  exact  and  proceed  as  follows  : 


SECOND    METHOD    OF    FINDING    H.   C.   F.         225 

11 


11^2__e^_5  )  143«2_264^  +  121  (  13 
143^--  ISa-  65 
-186a +  186 

Before  using  this  remainder  as  a  divisor  we  will  take 

out  the  factor  —186,  leaving  a—1,  and  then  proceed  as 

follows : 

a-1  )  11^2-  6a-5  (  ll«  +  5 

11^2-11^ 

5a— 5 

5«— 5 

Since  this  division  is  exact,  we  conclude  that  «— 1  is 
the  H.  C.  F.  of  the  expressions  we  started  with.  This 
is  an  unusually  hard  example,  and  the  student  who  can 
follow  this  will  not  find  any  trouble  with  any  example 
given. 

Examples. 

Find  the  H.  C.  F.  of  the  following  expressions  : 

27.  5x'--{-4x—l  and  20.:r2  4-2U-— 5. 

28.  2jt:a-4;i;2-13;t:-7  and  6x^-llx'--S1x-20. 

29.  x^-i-4x^-5x-20  and.r3  +  6ji:2-5j»;-30. 

30.  2;«;3__8^2  4.8^  and  Sx^-Qx^-dx-'  +  lSx. 

31.  3a2-22a-15  and  5a*+a^-'54a^-\-lSa. 

32.  x^—2x'^—x-r2  and  x^—Qx'^-j-llx—Q. 

33.  Sxjy{x^-x''-\-x-{-d)  and4j^/2(;t;4_^^3_3^2_^_^2). 

34.  x^-2x'^-\-2x-l  andx^-Sx^-\-2x^--^x-l. 

35.  x^  —Sxy^ —Sy^  and  x^—Ax^jz-^dj/^. 

36.  x^-Sx+Sandx^-{-Sx^+x-^S. 


226  HARDER    FACTORS,   MULTIPLES,   ETC. 

37.  x^-\-10x''-{-S3x+SQ  and  x^+dx^-{-2Sx+15. 

38.  x'<'-x''-4:X+4.and2x^-x^-6x+S. 

39.  2x^—5x'^—x+6  and  4x^-x^  —  llx-Q. 

40.  a^+bab-^W"  and  ^s4-4«'^+5a^2_p23^ 

41.  4:fn^—Sm-\-l  and  8/;z^  +  ;;e— 1. 

42.  2«3  4.^2_2^_6  and  Qa^-a'^-Ua  +  S. 

43.  8«3-8«2_4^_3  and  2«4  +  3«3_3^2_7^_3^ 

44.  x^—x'^—x—land  2x^+x'^—2x+l. 

45.  2jt:^^+;t:2 +2^-12  and  2jr3-7;t:2  4-14^-12. 

46.  «4_|.67^2^66  and  «4  4-2^''^+2«2+2«+l. 

47.  28^2  +  37^-21  and  35«2+62«-33. 

48.  7;;^^-13w2  +  34;;^-72and  7m'^-6m'^-\-S5m-S6. 

49.  2x^-Sax^-  —7a^x-{-4:a^  and  6x^  —lax'^  —Aa^x+Sa^ 

50.  2;»;3_^5:r2_)/+2^^2_^3  ^nd  dx^+2x''y+xy-{-2j\ 
51    2;*;3-6-r2-2^+6  and  3;i;4-9;t:24-6. 

52.  x^-x^—7x'^-\-x-{-6  and  ;t-*+^'*-7^2_^+6. 

53.  3^=^-4jt:2-;t;-14and  Qx^-lW-lOx-^l, 

54.  ^5_^3_^^_i  and  Jt-7— ;t:6— ;t:4  +  l. 

55.  10;t-=^+25^.r2— 5^3  and  4x--{-9ax'^—2a'-x—a^. 

56.  3;»;4-8;r-V  +  5jt:2y2_2.rjj/3  and  9x^  +  2x''y-\-y\ 

208.  When  we  wish  to  find  the  H.  C.  F.  of  more  than 
two  expressions,  we  first  find  the  H.  C.  F.  of  any  two  of 
them  and  then  find  the  H.  C.  F.  of  this  result  and  the 
third  expression,  and  so  on  until  all  the  expressions  are 
used.  The  final  result  is  the  H.  C.  F.  of  all  the  given 
expressions. 


FIRST    METHOD   OF    FINDING  L.   C.   M.  22/ 

Examples. 

Find  the  H.  C.  F.  of  the  following  expressions : 

57.  2x''  +  Sx-5,     Sx^-x-2,  and  2x''-{-x-S. 

58.  a^-Sa-2,     2a^-\-Sa^-l,  and  a^-\-l. 

59.  nCa^-d^),     10(«6-/^«),  and  S(iaH-ab^). 

60.  x''-x-12,     x''-j-(jx  +  8,  and  x^-4x^-x-{-4. 

61.  x^—6x^-^nx—Q,     x^-\-4x'^-{-x—(),     x^  —  Zx-k-2. 

62.  x^-lx''^-\4x-%,  x^-^x''-\-hx-\-\2,  ;f2-5j»;H-4. 

63.  x^-\-2x''-—x—2,     x'^+x^—x—l,  and  x^—x^. 

64.  .^-*+^J^^,     x"'v+y^,  and  x'^-{-x'^y'^ +  v^. 

65.  ;r»+3A-2-jr-3,     ;»;5-^-^  and  ji;2_3^+2. 

67.  x3  4.2;c2— ;»;— 2,     ;i:'+;»:4,  and  x'^-\-Ax-{-'6. 

68.  ;r2-2;»r--3,     ;»;2-7^+12,  and  x^+x''-%x-^. 

69.  ;i:-^  +  3;»r2_^_3^     ;»;4_2;r2-;t:+2,  and  x'^-x"^. 

70.  .r^+5;r-— ;tr— 5,     ;»;^— ^tr^,  and  x^^x^+x—1. 

EXERCISE  97.* 

L.  C.  M.  OF  Expressions  that  Can  be  Factored 

209.  The  lowest  common  multiple  of  two  or  more  ex- 
pressions has  already  been  defined  (see  Art.   S^)  and  a 

*  The  order  of  subjects  given  in  this  and  the  following  six  exercises  is  the 
one  usually  given  in  text  books  on  Algebra,  but  some  authors,  notably  Hall 
and  Knight  of  England,  prefer  to  take  up  the  subjects  after  H.C.F.  in  the  fol- 
lowing order :  I,  Fractions  Reduced  to  Lowest  Terms  ;  II,  Multiplication  of 
Fractions  ;  III,  Division  of  Fractions  ;  IV,  I^.  C.  M.;  V,  Addition  of  Fractions  ; 
VI,  Subtraction  of  Fractions.  This  order  has  the  advantage  of  wjzw.^the  H.C.F. 
immediately  after  it  is  treated,  and  also  using  the  L.C.M.  immediately  after  it 
is  treated.  At  the  option  01  the  teacher  the  order  here  spoken  ol  may  be  sub- 
stituted for  the  order  gfiven  in  the  text. 


228  HARDER    FACTORS,   MULTIPLES,   ETC, 

method  of  finding  the  L.  C.  M.  of  monomials  has  been 
given.  We  now  take  up  the  case  of  the  L.  C.  M.  oi 
polynomials.  First  we  will  consider  the  case  oi  ex- 
pressions which  can  be  readih^  lactored.  The  method  ol 
finding  the  ly.  C.  M.  in  this  case  is  exactly  like  that  given 
for  monomials,  viz.: 

Resolve  each  expression  iJito  its  prime  factors  a7id  form  a 
jyroduct  in  which  each  of  thetn  occurs  as  inany  times  as  ii 
occurs  in  that  07ie  of  the  given  expressio7is  in  zvhich  it  occurs 
the  greatest  number  of  times. 

Examples. 
Find  the  L.  C.  M.  of  the  following  expressions  : 

1.  x^  —  1  and  jr^  — 1. 

2.  x^  —  1,     x"^  —  !,  and  x—1. 

3.  ji-2  +  7jf+12,     x^'+Sx+Id,  and   x^'-j-Sx-j-G. 

4.  x'^-x-(^,     x'^+x-'l,  and  x'^-'ix—VZ. 

^    x-ij^x-AI,     x2-lLr-f  30,  and  .^2+2:^-35. 
b.  Aabia'^-Zab+lb'')  and  ba\a'' -\-ab-W). 

7.  x'^y—xy'^,     Zx^x—yY,  and  4y(x—y)^, 

8.  x—y,     x-\-y,     x'^—y^,  and  x'^—y'^. 

9.  a—b,     a  +  b,     a'^  —  b'^,  and   a'^-{-b^. 

10.  x^—4x'^-\-i^x,     x'^+x-'  —  VIx'-,     x''+Sx^  —  4x^. 

11.  20(.r'-^-l),     24(jr2-x-2),  and  16(.r2+:r-2). 

12.  x'^  —  7x-^6,     x'^  —  ox—6,  and  x^  —  1. 

13.  2{x-yy,     8(x-j/),     3(;r+^),  and  GCr^+j^^). 

14.  x'^-\'7x-}-Q,     x^--^Qx—7,  and  x^  —  6x—7. 

15.  .:r2-;i;-42,     x''-9x  +  U,  and   j»;2+4j»;-12. 


SECOND    METHOD    OF    FINDING    L.   C.   M.         229 

i6.  5(«2_2«^),     10(«/^+2<^2-)    and  15(aH^-U^), 

17.  x^-5x-{-6,     .r^-S,  and  x^-21x''. 

18.  x''+x-2,     x^-\-S,  and  x^-x\ 

19.  (2;;24-2)2,     (w  +  l)^     5;«  +  5,  and  ;^^2_i^ 

20.  (^— 1)^     (2;tr— 2)2,   and  ;i:*-2;»;2  +  l. 

EXERCISE  98 

L.  C.  M.  OF  Expressions  Not  Easily  Factored. 

210.  When  we  wish  to  find  the  L.  C.  M.  of  two  ex- 
pressions not  easily  factored,  we  first  find  the  H.  C.  F.  oi 
the  two  expressions  by  the  method  of  exercise  96.  This 
H.  C.  F.  is  of  course  07ie  factor  of  each  of  the  two  given 
expressions,  and  the  other  factor  is  obtained  by  dividing 
each  expression  in  turn  by  the  H.  C.  F.  In  this  way  we 
factor  each  of  the  given  expressions,  and  then,  when  the 
factors  are  known,  we  proceed  as  in  the  previous  exercise 
to  find  the  L.  C.  M.  of  the  two  expressions. 

Let  us  find  the  L.  C.  M.  of 

6:r3-lLr-'r+2y  and  ^x^ —llxy'^ —'^y^ . 

Wq  Jirsi  find  the  H.  C.  F.  of  these  two  expressions  as 

follows  : 

6;t:3-lLrV  +   2y^ 

3     ■ 

9ji;3— 22;9/2_s^3  )  iSx''-ZZx''y  -f  6^  (  2 

18^^ _44^^2_ie>/g 

— llj  )  -SSx'^y-hUxy''  +  22y^ 

3:^2   —  4xy  —  2y'^ 

Sx^—4xy'-2y'^  )  9x^  —22xy''-Sy^  (  Sx+4y 

9x'^—12x'^y—  6xy'^ 

12x^y — 16xy'^ — 8y* 
12x^y—Uxy'^—Sy^ 


230  HARDER    FACTORS,   MULTIPLES,   ETC. 

From  this  work  we  see  that  Sx'^—4xj/—2y^  is  the 
H.  C.  F.  of  the  two  given  expressions,  and  if  we  divide 
each  expression  in  turn  by  this  H.  C.  F.  we  obtain  the 
other  factors  as  follows  : 

Sx''-ixy-2y  )  Qx^-llx'^y  H-2j/»  (  2x-y 

—  Sx'^y-\-4xy'^  -^2y^ 

—  Zx'^y-i-4xy'^-^2y^ 

Sx^—4xy—2y'^  )  9x^  —22xy'^—Sy^  (  Sx+4y 

9x^—12x'^y—  Qxy'^ 

+  12xy—16xy'^—Sy^ 

■i-12x^y—26xy'^—Sy^ 

From  the  first  of  these  two  divisions  we  have 

Qx^  -llx''y-\-2y^  =  (Sx''  -4xy-2y'')(2^-j'). 
and  from  the  second  of  these  divisions  we  have 

9x^-22xy''-8y^  =  (Sx^-4xy—2y'')(Sx-\-4y). 

Now  we  have  the  two  given  expressions  factored,  and 

from  these  factors  we  can  readily  write  down  the  L<.  C.  M. 

of  the  two  given  expressions.     Plainly,  this  I^.  C.  M.  is 

{Zx''-4xy-2y'''){2x-y')(^Zx-^4y). 

Examples. 

Find  the  L.  C.  M.  of  the  following  expressions  : 

1.  x^-\-8x''-^V^x-\-\2  and  Jt;^4-7:r2+7jr— 15. 

2.  .;»;-^  +  8;i;2  +  19.r  +  12  2.n^  x^ -{■Zx'^—4x—\2. 

3.  j»;3 +70.-2+7.^-15  andj»;3  +  3.r2-4.r-12. 

4.  5.r24-ll..;t+2  and  15.;i:4+48ji;3+9ji:2. 

5.  4.r'^-10j»r2+4.r4-2and  Zx^-2x^-Zx-V2. 


FRACTIONS    REDUCED    TO    LOWEST    TERMS.      23 1 

6.  (yx^'-^-llxy-^Ay^-  and  4x^-Sxy-5y^. 

7.  x^+x^—6x-hSandx^  —  Sx'^-i-Sx—l. 

8.  x^+xy'^-i-2j'^  and  x^-\-x'^y-\-4y^. 

9.  x'  +  2x^+x-''+8x''-i-Wx+Sandx'-4x^+x'^-4. 
10.  x^  —  ix'^+x^—A  and  x^ +2x'^—ox—12. 

211.  If  we  wish  the  L.  C.  M.  of  more  than  two  ex- 
pressions, we  first  find  the  L.  C.  M.  of  any  two  of  them 
and  then  the  L-  C.  M.  of  this  result  and  the  third  ex- 
pression, and  so  on  until  all  the  expressions  are  used ; 
the  result  is  the  L,,  C.  M.  of  all  the  given  expressions. 

Examples. 

Find  the  L,.  C.  M.  of  the  following  expressions  : 

11.  2;»;2-f2;t:— 1,     3.;<;'^— 4r-f  1,  and  2x^—Sx-\-l. 

12.  x^+2x-+9,     x^-Sx-{-S,  and  x^-Sx+1. 

13.  .r2_2;t_2,     x^-4x~+S,  and  x^ -Sx'^ -\-2. 

14.  9x^-j-2x-^l,     3;r3— 8.:r2  +  l,  and  x'^—Sx-\-l. 

15.  x'^—Sx-\-2,     A-3— 6.t:2  +  lU— 6,  and  x^—5x-{-6. 

EXERCISE  99. 

Fractions  Reduced  to  Lowest  Terms. 

212.  The  general  properties  of  fractions  have  already- 
been  given,  and  all  the  operations  connected  with  frac- 
tions have  been  given  in  the  case  that  both  numerator 
and  denominator  are  monomials.  It  only  remains  now 
to  apply  the  general  properties  and  methods  to  fractions 
in  which  the  numerator  and  denominator  may  be  poly- 
nomials as  well  as  monomials.     The  first  thing  v.e  have 


232  HARDER    FACTORS,   MULTIPLES,   ETC. 

to  consider  is  the  reduction  of  fractions  to  their  lowest 
terms.  This  is  done  exactly  as  before,  b}^  dividing  both 
numerator  and  denominator  by  the  H.  C.  F.  of  the 
numerator  and  denominator. 

Examples. 

Reduce  the  following  fractions  to  their  lowest  terms  : 


2a''-^al?-P  4^4  4- 11^- +  25 

x"^ -\- ax -\- ex  +  ac  ^a^ -\-2a'^  —  15a—  6 

x'^ -]- dx -j- ex  +  dc  .7^^  — 4^2—21^  +  12' 

(Sx-2yy  -{2x-i-2j'y  ^-"^ +6^- +  11^  +  6 

£3__39^70  36^^-18^2  +  1 

•    x'^-dx-lO'  ^  '   30a=^-19«2^r 

{a-^by-i^c+dy  ^-5-10^2  _^  26^-8 

7-  7-T-?r^ — 7irT^T^'  17. 


{a  +  cy  —  {b+dy'  ''  ^3-9^2^23^-12' 

3.^=^-6^1:2 +.r-2  ^  8r^-10^2_i6^_3 

18. 


x'^-1x-\-^  '  6^4_22,-3_^31^2_23^_7 

x^+x^xj-B  2a^-9a'"-14^  +  3 

^'       x^-x-Q     '  ^^'   Sa-'~Ua^-da-i-2' 

x^+a^  l  +  2«— 3^2 

10.     -TT^^ .— TT.  20. 


x'^-\-2ax+a^'  '   l—Sa—2a^+4:a^' 


ADDITION    OF    FRACTIONS.  233 

EXERCISE  100. 

Addition  of  Fractions. 

213.  In  exercise  08  we  considered  the  subject  of  the 
addition  of  fractions  in  which  the  numerator  and  denom- 
inator were  each  monomials.  We  now  extend  the  subject 
there  treated  to  the  case  in  which  the  numerator  or  de- 
nominator or  both  are  polynomials.  Of  course  the  method 
is  the  same  here  as  in  exercise  58,  viz. :  first,  reduce  the 
fractions  to  a  common  de^iominator,  and  then  add  the 
numerators.  Here,  as  before,  we  take  the  L.  C.  M.  of 
the  given  denominators  for  a  common  denominator. 

Examples. 

Add  the  following  fractions  : 

I.   — -, —  and  —zrz. — .  3-  ^   and 


6  12  ^    x—\  X-—1 

4a— d        ,  ba—d  w^  ,        nr 

2.  — r-^  and     .,    „  .  4.  7 — — ^o  and 


Sa-4      5a-6        ^    7a-S 

5.  -TTX'    -:ri:T'  ^^^ 


a-\-d'      a-b'  a'^-b'^' 

6.      „  .  ,^     and 


x''  +  2x  x''-2x 

x—y      x-\-y  x--\-y^ 

„    u-\-v-\-w       u-\-v-{-w         .         1 

8.  ; ,     ,  and  — — 

n-\-v  u—v  U-  — 

n  r  .         \ 

9.  — ; — , ,  and  —:^. — ^. 

'    n-\-r       71— r  w-fr- 

a-^b          a—b  .       ab 

10.      „  .   ,^,     -r^ T^,  and 


^24.^2'     a-i-b"''  «=^-f/^»* 


234 


HARDER    FACTORS,   MULTIPLES,   ETC. 


II 
12 
13 
14 
15 
16 

18 

19 
20 


and 


x(x—a) 


x(x—by 

x+y  ,       1 

■^    ,  and 


x'^-y'' 


1 


a  +  b-\-c      a-^b 
x+1  i 


x—y 
and 


a-\-c 


and 


3.^4-4 

^2+5^+6'       (ji:  +  2)2'  —  (^  +  3)2* 

2x+5  ,         2jc-f7 


and 


y  .      1 


X 


,  and 


;r  ^         ;r2+^-2 

7  and 


^"— ^ 


x-\-a      x-\-b 

x^  -\-a 
x'^+a^x'^+a^'  x^-\-ax-\-a'-' 


x"^  -\- ax -j- at 
x-\-a 


and 


x—a 


x^—  ax-{-a^ 


be      b'^ —  ac 


be 


ae 


and 
1 


ab 


ab 


1  1  ,        1 


EXERCISE  101 


Subtraction  of  Fractions. 


214.  In  exercise  59  we  considered  the  subject  of 
subtraction  of  fractions  in  which  both  numerator  and 
denominator  were  monomials,  and  it  only  remains  now 
to  take  up  fractions  in  which  numerator  or  denominator 
or  both  are  polynomials.  The  method  is  of  course  the 
same  as  in  exercise  59,  viz.:  reduee  the  fraetioiis  to  a 
common  denominator  and  the7i  subtract  the  mtmerator  oj 
the  subtrahend  from,  that  of  the  7ninucnd. 


SUBTRACTION    OF    FRACTIONS.  235 

Examples. 

2;t:+l      ,     3.r+2 

1.  From  — ^ —  take  — 7 — . 

3  4 

^         ax-i-d     .     cx-^b 

2.  From  take . 

c  a 

3.  From  ^^  take  ^^. 

x—y  x-\-y 

4.  From  — ^ — ~.  take  .,. 

5.  From     ..    ^^  take  - „-, — — — ;?. 

6.  From     „  .  r — r-77  take 


^y      take  ^>t:^' 
r'^  -\-y^  x^  —y' 


7-  i^^^o"^ -;:3^-;;3  take    ^,_      . 


o    ^         x'^+xy-\-y'^      ,          jr^ — j/^ 
8.  From       ^  -^V-  take  ^ ^r. 

'            x'^+xy  •\-y^      ,     jr^— .rr  +  y^ 
9    From  —      r         -  take ^'-L^-. 

10.  From  7 TT re  take 


(^x-aXa-d)  {x-b){a-by 

II.  From  —X——, — — r^ — : — 7  take 


x'^-\-(a-\-b)x-\-ab  x'^-\-(a-^c)x+ac 

In  the  following  examples,  perform  the  additions  and 
subtractions  indicated  and  express  the  result  as  a  single 
fraction  in  its  lowest  terms  : 

1.1  1 

12. 


13. 


x+\  '  x-\-2     x-^^' 

x+\  ,         x-^2  x-^B 


Cr+2)(;^+3)  '  (x+lXv+S)     ix-\-lXx-f-2y 


^6  HARDER    FACTORS,    MULTIPLES,   ETC. 

1  2 3_ 

Zb     5c^ 


( a      b      c\       fa      6b     bc\ 

'5-  b+s-lj  +  U-T-e-j- 


^    a-\-b  ^  b—a        Aab 
i6.  ,4- 


17- 


a — b  '  a  +  b     a'^  —  b'^' 

2x-\-l 3x  +  2 Ax-\-Z 

{x-Vt^x"^^)     (;t--2)(x^     (x-Z)(^x~^y 


a         a-\-\          a         a"^ — a-\-l 
lo.   '--{■ 


19- 


a — 1      a'^ -\-l      a-\-l        a'^  —  1 

3<2— 6  4«— 5  a—1 

-7«  +  12~a2_8^_15— ^2_9^^20' 


2^4-1        3^  +  2 2_         3 


EXERCISE  102. 

Multiplication  of  Fractions. 


215.  In  exercise  60  we  considered  the  multiplication 
of  fractions  in  which  both  numerator  and  denominator 
were  monomials,  and  it  only  remains  here  to  extend  the 
operation  to  the  case  where  the  numerator  or  denominator 
or  both  are  polynomials.  Of  course  the  method  used 
here  is  the  same  as  in  exercise  60,  viz. :  multiply  ah  the 
numerators  together  fof"  a  7iew  riuTnerator  and  all  the  ae- 
nominators  together  Jor  a  7iew  denominator.  The  result 
should  be  reduced  to  its  lowest  terms  if  it  is  not  in  its 
lowest  terms  already. 

216.  Instead  of  actually  performing  the  multiplications 
it  will  frequently  be  best  to  iiidicate  them  by  using  paren- 
theses, for  sometimes  in  the  result   the   numerator  and 


MULTIPLICATION    OF    FRACTIONS.  237 

denominator  will  contain  a  common  factor,  which  can  be 

struck  out  and  thus  save  the  trouble  of  multiplying  by 

these  factors.     For  example,  if  we  wish  the  product  of 

a-^x        b  ,  c—x 

—7--,    — — ,  and  — ; — , 

we  can  write  the  product  thus : 

d(a-{-x')(c—x) 
dic-i-x)(ia+xy 
Here  the  numerator  and  denominator  contain  the  com- 
mon  factor   d(a-{-x'),   which   being  rejected   from   both 
numerator  and  denominator  leaves  the  result  simply 

c—x 
c+x' 

Examples. 

Find  the  product  of  the  following  fractions : 

b-j-c       ,  d—c  a—b      ^  a"" -\-ab-\-b'^ 

1.  — —  and  .  3.  -.and  -^^ — ,  ,    ,„. 

x-\-c  x—c  c—d         c--\-  cd-\-d^ 

x-\-\       ^x'^-x^-X  a'^b'^  +  ^ab       _,  2a-f  1 

2.  — — -  and  —2 T-:r.        4-   —j-^ — r~  ^^^     ,  ,  o- 

x^-\-xy-\-y'^       ,  x"^ — xv-\-y^ 

5.   2—^-  ^^^ Tl^~' 

x^—y  x^-\-y^ 

^    «2_i21       ^   a  +  2 

6.  — 5 — —  and  — —- : . 

«2_4  a-f-11 

7-  4:r2~'9  2jr2  +  llj»;-f5* 

_      Ux-—7x         ,  x'^-^2x 

^'  T2:^^M^24^  ^^^  "2:^3r- 

Sa'^—4ab       Uja—b)  a-j-b 

^'    l{a  +  b)  '     3^M^3^'  '^b' 

10.   -a — FT-T-r-r^,     ,  and 


«2_2«^-f^2'  ^         >    "-    ^_|_^' 


238  HARDER    FACTORS,   MULTIPLES,   ETC. 

"•  -TTTi.     -T^-Zh^  and 


12.  ,     5,  and 


1  +  ^  '     a — <2^'  1 — a 

a^—x^  ^2 — ^2 

^3-    -^-j — ^  and  — — ^ ; — -. 

a^-\-x^  a-  —  2ax-{-x^ 

^^'       2ab    '     a^^b^-'  Aa{a-\-b)   * 

a^-4^^  ab  +  W 

^^'   a^a  +  ^b-)  a(^a-2by 

^    a^  —  la'^x'^-Vx^       ^  a'^+x'^ 

10.  -—-- —  and     ., 

a'^x-i'ax-^  a^—x-' 

«^+3«-/^  +  3<2<^2_j_^3       a—b  ,     a 

^7.  TV— t;.^ 5    ,0  ,  ,i>  and 


^2_^2  '  a'^-\-aV  a  +  b' 

o    x^'-dx+'IO    x--lSx-{-42        ,    ^2 

lo. ^^ — ^r ,  ~ — ,  and ^. 

x^—bx  x^—Qx  x—1 

x^-xy+r^  £±?:,  and  to^. 
x^—y^        x—y  x^-\-y^ 

20. T— ,  ^ — - — ,  and 


•2  —  9     '         jr2-4^+3' 

EXERCISE  103. 

Division  of  Fractions. 

217.  In  exercise  61  we  considered  the  subject  O-  the 
division  of  fractions  in  which  both  numerator  and  denom- 
inator were  monomials,  and  we  now  have  to  extend  the 
subject  to  the  case  where  the  numerator  or  denominator 
or  both  are  polynomials.  The  method  used  here  is  of 
course  the  same  as  in  exercise  61,  viz.:  invert  the  terms 
of  the  divisor  and  proceed  as  i7i  multiplicatiori. 


DIVISION    OF    FRACTIONS.  239 

Examples. 

1.  Divide by . 

a—x  X 

_.    .,    x-  —  lox+-^'l  .      x—7 

2.  Divide :; — by  — ^-. 

x'—ox  x^ 

x'-dx+20.  x-'-hx 

3-   Divide  — ^^z:^—  by  -:2Zrr3^4:42- 

4.  Divide  -^T— — z  by  -ly — ^ ;-— -^. 

5.  Divide  ;^-^7;7^Yo  ^^'  :^+6]^T^- 

6.  Divide  ^^3-^^^^-^  by  ^3-^. 

^.    .,    «-— 8a  — 4,      a^— 16 

7.  Divide  — ^^ — 3 by  — ;; . 

o     X....     x''-V^ax-'^-\-?>a''^x^a^  .      x'^^-a'^ 

8.  Divide  w~7-.^, — r-Uy "  by 


9.  Divide 1-- by ^^ 

10.   Divide r — ' by 


11.  Divide --3^^- by  ^^-^. 

.    .^     Cr+1)-  ^     (^-1)' 

12.  Divide -^^  by -^^^. 

13.  Divide  ^^-,  by  --. 

14.  Divide  -  ^.+2a+r-:r«^~  ^^  -^^F+r- 


240  HARDER    FACTORS,    MULTIPLES,   ETC. 


1—^2  --y  (l-\-a) 


1' 


1 6.  Divide 7, — J7, by —=—. 

a-'  —  b-  a-\-b 

^.    .,     a-'  —  b'''  -\-a  —  b  .      a  —  b 

17.  Divide  — — ^,-— --7  by  — -,. 

a''  —  b--\-a-\-ba-\-b 

18.  Divide  ^ ^^-^ r,-  by =^— — . 

19.  Divide— ^^-^,— by— ^^^. 

20.  Divide —  by 


X — 1  \-\-ax' 

EXERCISE  104. 

Miscellaneous  Fractions. 

218.  We  may  take  an  integral  expression  (/.  e.,  a 
monomial  or  polynomial  in  which  there  is  no  fraction 
involved,)  along  with  one  or  more  fractions,  giving  us  a 
form  partly  integral  and  partly  fractional.  Such  expres- 
sions are  sometimes  called  Mixed  Expressions  or  num- 
bers. Mixed  expressions  maj^  always  be  reduced  to  the 
form  of  fractions  by  writing  the  integral  part  in  the  form 
of  a  fraction  with  a  denominator  1 ,  and  then  performing 
the  indicated  operations  as  before  explained.  For  ex- 
ample, suppose  we  wish  to  express 

in  the  form  of  a  fraction.    We  write  the  expression  thus : 

-^1  J^yl  ^y 


+ 


X"' 


MISCELLANEOUS   FRACTIONS.  241 

Reducing  to  a  common  denominator  and  adding  in  the 

usual  way,  we  get 

x*—y^+xy 

X'—jy^ 
This  process  may  be  called  Reducing  Mixed  Expres- 
sions to  Fractions. 

Examples. 

Reduce  the  following  mixed  expressions  to  fractions : 

X 


I. 

X 

2. 

x'  +  l+l- 

3- 

x—a 

4. 

^     X    y 

5. 

a     0      c 

6.  x^+x+1 

7.  a-^-d+c^ 


1 


a-^b+c 
b^ 


0.  x'^—ax+a'^-^ 
9.  (x-i-ay-h 
10.  d:;c4-^ 


1 


x-\-a 
1 


ax— by 


219.  When  the  given  mixed  expression  has  only  one 

xy 
fraction,  as  in  the  example  x--hy'^-\ — tt^ — =,  we  reduced 
^  -^        x-—y^ 

to  a  common  denominator  by  multiplying  numerator  and 

denominator  of  — y^hy  x'^—y'^,  the  denominator  of  the 

fractional  part  of  the  given  expression.  This  amounts  to 
multiplying  the  integral  part  by  the  given  denominator 
and  acjding  the  namerator  exactly  as  in  Arithmetic. 

Since  the  fraction  was  produced  from  the  mixed  ex- 
pression by  multiplying  the  denominator  by  the  integral 
part  and  adding  the  numerator,  it  follows  that  to  reverse 
this  process  and  go  back  from  the  fraction  to  the  mixed 

16 


242  HARDER    FACTORS,   MULTIPLES,   ETC. 

expression  we  would  divide  the  riumerator  by  the  denom- 
inator as  far  as  possible  and  ivrite  the  quotient  for  the  inte- 
gral part  and  the  7'emainder  over  the  denominator  for  the 
.  fractional  part,  which  again  is  exactly  as  in  Arithmetic. 

Examples. 

Change  the  following  fractions  to  mixed  expressions  : 

"•  x'-^y  '  ^^'  iO        "• 

x--hax-^a'^                               \2a'^J^Aa—bc 
"•         ,.^a       •  ^4.  4^ . 

x'^-^a'^x'^-^a^-^x  +  a 


15. 


16. 


X'-^ax-j-a'^ 
x^  -hSax''  -j~Sa'^x-\-a^  -ha^ 


x-j-a 

x-—y"+z'^                               ax  +  by+c 
10. .  20. . 

x-^y  x+y 

220.  By  combinations  or  repeated  applications  of  the 
preceding  processes  we  are  able  to  deal  with  more  com- 
plicated cases  than  have  yet  been  given.  For  example, 
let  us  take  the  fraction 

x-\-a    x—a 

X — a     x-j-a 

x-\-a     x—a 

x—a     x-\-a 

The  numerator  =^-'?_£---'^=^f±^^f=f2!. 
X—a     x-\-a  x^—a^ 

^  (^x"-  -V2ax^a'^')-{x''-1ax-\-a'^) 

x-'-a"-' 

4  ax 


MISCELLANEOUS    FRACTIONS.  243 


The  denominator  = 1 ; — , 

x—a     x-\-a 

{_x-^ay-{-{x-ay 


x^—a^ 
(x^+2ax-\-a^)-{-{x'^-2ax-\-a^') 


x'^^a^ 


x'-a'' 
Therefore  the  original  fraction 

^    4ax     ,  2(x^--a'-) 

x^—a^  '     x'^—a"^ 
__    4ax       .  x'^—a'^ 
'^x'-a'  ^  2{x''-j-a'y 
Aax  2a  X 


This  final  result  is  much  simpler  than  the  fraction  we 
started  with,  so  this  kind  of  work  may  be  called  Sim- 
plifying Complex  Fractions  or  Expressions  Involving 
Fractions. 

Examples. 

Simplify  the  following  expressions  : 

21.  ± 24.  £ £ ^. 


x-^Z-V-  a'^'-ab  +  b'' 

X  a-^b 

y     X 


X-\ q-  1— :j— —      1— :j 

x—\  l-hx  1—x 


244  HARDER   FACTORS,   MULTIPLES,  ETC. 

/.      ^^\      /-.     ab—b^-X  a*  a—b 


^3_^^3  ^2_^^^_|.^2  ^2_|_^^_^^2 


30.  -^-imT-iTX- 


or      (^  +  ^)^-  (^+^)^-  ^.  (^-^)2_  (^_^)2 

2  •   {a-\-cy-{b-^dy     {a-cy-ib-dy 

33     p 1-U '- 

^^'    \n—r     71— s)     7i'-  —  7i{r-\-s)-\-rs 

fa^—b^     a^^-b^\         4ab 

s=[G«)'-(H)']*[e-^)'+(H'] 


38.  i-2^-  40. 

a — o 

i-i 
1 — ^ 

39-   j^^  41- 


''    x'^  +  a"     ■  Va 

x) 

-.      '  +  \ 

«2      a;i:^;i:2 

1 

^      .(^-J^)^- 

-Axy 

1      1-1.1 

2  ' 

MISCELLANEOUS   FRACTIONS.  245 

(aj' — bx)  ^  -f  {ax  -\-  by)  ^ 

1 
43.  1 


;ir-l  + 


1+    " 


Fractions  like  this  are  called  Continued  Fractions. 
To  simplify  a  continued  fraction,  begin  at  the  lowest  part 
and  proceed  upward  step  by  step  as  follows  : 

■*"4r^~    4-x    ~4-Jc' 
Hence  the  original  fraction  may  be  written 


But 


4  —  a; 
1        4-aj 


4  4 

4-x 
Hence  the  original  fraction  may  be  written 

1 
1     4-  a; 

L  —  x    4a;— 4  4-4— ^     3aj 
But  x-l-f-^= ^ =— • 

1       4 

Hence  the  original  fraction  =.t-:=^7—. 

ox    oa? 

4~ 
X 

44.    1 45.    — ^—         46.    -— T 


246  HARDER    FACTORS,   MULTIPLES,  ETC. 

j,r^     o     a  a  a^ 

4u.   -_ X 7  H-  -s — 7^-. 

a-^     ^      a—o     a^     <?- 


<9. 


50. 


-1     x-\     x-\-Z    x^Z 

3  ""'".r-2        7        A^-f4 


x-\-2     x-i-2  '  x—2     x—2 


1+1    1 

X  X 


;i;  X 


221.  When  an  operation  is  performed  upon  a  polyno- 
mial, some  or  all  of  whose  terms  are  fractions,  we  natu- 
rally combine  all  the  terms  into  a  single  fraction,  and  t/ien 
perform  the  indicated  operation.  Sometimes,  however, 
the  operation  may  be  performed  without  thus  combining. 
For  instance,  if  we  wish  to  multiply 
a"^      a      1         a      1 

we  would  by  the  previous  process  combine  each  of  these 
expressions  into  a  single  fraction  as  folows: 


and 
Therefore, 


a     1_3^— 2 
2     3~     G     * 


V2"^'3'^4A2     3)"         12         ^ 


3«~2 

— n — > 


72 


MISCELLANEOUS   FRACTIONS.  247 

But  we  may  also  multiply  these  expressions  together  the 
same  as  integral  expressions  were  multiplied,  as  follows: 


2  ^3^4 

a      1 

2     3 

«3        ^2^ 

a"^     a 

1 

G"     9 

"12 

a^      a      a 

1 

T"^8      9" 

"12 

The  terms  of  this  result  may  be  combined  after  reduc- 
ing to  a  common  denominator,  72.      Comb-ning  terms, 

a^     a_a 1  _  18^''-+9^-8a-G^  18^=^+^-6 

4"^8     9     12""  72  ~         72 

which  is  the  same  as  was  reached  by  the  other  process. 

222.  As  another  illustration  let  us  divide 
^^_13a2     X        a__\ 

6       30   "^  4  ^  3     2 
The  work  may  be  arranged  as  follows: 

3     2>/  G        3G       ■*"4V2     3     2 

6       4" 


9      ^4 

G^4 
G^4 


248  HARDER   FACTORS,   MULTIPLES,  ETC. 

The  terms  of  this  quotient  may  be  combined,  giving 

6 ' 

which  is  the  same  result  as  would  be  obtained  by  re- 
ducing both  dividend  and  divisor  to  a  single  fraction  and 
proceeding  by  the  method  already  given  for  dividing  one 
fraction  by  another. 

Examples. 

By  the  method  used  in  these  two  illustrations  work  the 
following  examples  : 

51.  Multiply  .r^H — by  ;t: 

52.  Multiply  ~+^+l  by  |+i 

53.  Multiply  — +^— .  by  — -f  • 

a       0       ab         a      b 

54.  Divide -^--^+-+1-- by  2-3. 

/I      1\2  1      1 

55.  Divide  (-+^)    -lby--f--l. 

56.  Divide -X ^hy -r 

lb     a^         4      a 

57.  Divide  «3_fL-|-^_____  by  a+g- 

27  3 

58.  Divide  8a'^-f-  3  by  1a^ —  and  multiply  the  result 

1 
bya+-. 
a 

59.  Multiply  a;2 _  by  .r-f-  and  divide  the  result 

X  X 

,            1 
by  X 

X 


CHAPTER  XIV. 
QUADRATIC  EQUATIONS. 

EXERCISE  105. 

Preliminary  Tones. 

223.  The  Degree  of  a  Monomial  with  respect  to  any 
letter  or  letters  it  may  contain  is  the  sum  of  the  exponents 
of  the  letters  named;  unity  being  always  understood 
where  no  exponent  is  written.  Thus,  hab'^x^y^  is  of  the 
first  degree  with  respect  to  a,  of  the  second  degree  with 
respect  to  b,  of  the  third  degree  with  respect  to  x,  of  the 
fourth  degree  with  respect  to  y,  of  the  seventh  degree 
with  respect  to  x  and  y,  of  the  seventh  degree  with 
respect  to  a,  b,  and  y,  etc. 

1.  What  is  the  degree  of  ^a'^b^x'^y^  with  respect  to  a  ? 
What  with  respect  to  .^?  What  with  respect  to  _y  ? 
What  with  respect  to  a,  b  and  x>  What  with  respect  to 
x  andjj/? 

2.  What  is  the  degree  of  oa'^x^y^  with  respect  to.r? 
What  with  respect  to^  ?  What  with  respect  to  x  and  y  ? 
What  with  respect  to  a  and  x  ?  What  with  respect  to 
a,  X  and  r? 

224.  When  the  degree  of  a  monomial  is  spoken  of 
without  speciiying  the  letters  with  respect  to  which  the 
degree  is  taken,  it  is  usually  understood  to  mean  the  de- 
gree with  respect  to  all  the  letters  it  contains,  and  is  then 
equal  to  the  number  of  literal  prime  factors,  or  what  is 
the  same  thing,  the  sum  of  all  the  exponents  of  the  let- 
ters in  the  expression. 


2  50  QUADRATIC   EQUATIONS. 

225.  The  Degree  of  a  Polynomial  with  respect  to  any 
letter  or  letters  it  may  cojitain  is  the  degree  of  that  one  of 
its  terms  whose  degree  with  respect  to  the  specified  letters 
is  highest.     Thus, 

a^x'^  -\-abc^x-\-e'^x^y^ 

is  of  the  second  degree  with  respect  to  a,  because  the  first 
term  is  of  the  second  degree  respect  to  a,  and  neither  of 
the  other  terms  are  of  so  high  degree  with  respect  to  a. 

The  same  expression  is  of  the  fourth  degree  with  re- 
spect to  Xy  because  the  third  term  is  of  the  fourth  degree 
with  respect  to  x  and  neither  of  the  other  terms  are  of  so 
high  degree  with  respect  to  x. 

3.  What  is  the  degree  ax''-y-\-bxy'^-\-x'^y'^  with  respect 
to  ^  ?  What  with  respect  to  _>/  ?  What  with  respect  to 
x  and  y  ?  What  with  respect  to  «  ?  What  with  respect 
to  a  and  b  ? 

4.  What  is  the  degree  of  x'^y-\-xy^ -\-x^  with  respect 
to  ;r  ?  What  with  respect  to  ^  ?  What  with  respect  to 
x  and  y  ? 

Ans.  5,  for  the  degree  with  respect  to  x  alone  is  5,  and  the  term 
that  determines  the  degree  has  no  y,  so  it  leaves  the  degree  5. 

5.  What  is  the  degree  oi  x'^ax'^y-Vbxy'^  -\-aby''''  with  re- 
spect to  ;tr  ?  What  with  respect  tojK?  What  with  respect 
to  X  andjj/? 

6.  What  is  the  degree  of  a'^ bx ■\- b''" xy^ -{- cxy"^  with  re- 
spect to  «  ?  What  with  respect  to  ;i:  ?  What  with  respect 
to  J?  What  with  respect  a  and  jr?  What  with  respect 
to  a,  b,  Cy  X  and  y. 

226.  When  the  degree  of  a  polynomial  is  spoken  of 
without  specifying  the  letters  with  respect  to  which  the  de- 
gree is  taken,  it  is  usually  understood  to  mean  the  degree 


PRELIMINARY    TOPICS.  2$  I 

with  respect  to  all  the  letters  it  contains,  and  is  then 
equal  to  the  number  of  literal  prime  factors  in  that  term 
which  contains  the  greatest  number  of  such  literal  prime 
factors, 

227.  The  Degree  of  an  Equation  is  its  degree  with 
respect  io  the  2inknown  numbers  or  quantities,  i.  e.,  it  is  the 
degree  of  that  one  of  its  terms  whose  degree  with  respect 
to  the  unknown  number  is  the  highest. 

Remember  that  the  last  letters  of  the  alphabet  are  used  to  stand 
for  unknown  numbers. 

7.  What  is  the  degree  of  <a!^;»r+^'*_>/2=4?     Oi ax'^-\-bxy 
^cy^==ci}     Of  ax''  +  dx-{-c=0} 

228.  A  Quadratic  Equation  is  only  another  name 
for  an  equation  of  the  second  degree.  In  this  chapter  we 
deal  only  with  equations  with  07ie  unknown  number. 

229.  Quadratic  Equations  are  divided  into  two  classes: 
Pure  or  Incomplete  and  Affected  or  Complete. 

230.  A  Pure  or  Incomplete  quadratic  equation  is  one 
which  contains  the  second  but  not  the  first  power  of  the 

unknown  number,  as  ojir-  =  12  and  — = — =2. 

5 

231.  An  Affected  or  Complete  quadratic  equation  is 
one  which  contains  both  the  second  and  first  powers  of 

X         X 

the  unknown  number,  as  ox'^  +  4x=Sd  and  —-4-— =3. 

o       4 

232.  A  Root  of  an  equation  is  any  number  which 
substituted  for  the  unknown  number  will  satisfy  the 
equation,  z.  e.,  will  cause  the  equation  to  be  true.  For 
example,  2  is  a  root  of  the  equation  x'^-\-x=Qi,  for  if  2  be 


252  QUADRATIC    EQUATIONS. 

written  in  place  of  x  we  get  2^+2=6,  or  4  +  2=G,  which 
is  true.  Again,  —3  is  also  a  root  of  x'^-\-x=G,  for  if  — 3 
be  written  in  place  of  .^twe  get  (—3)2—3=6,  or  9—3=6, 
which  is  true. 

Although  we  deal  in  this  chapter  only  with  equations 
of  the  second  degree,  still  this  definition  of  root  will  hold 
good  for  an  equation  of  any  degree  whatever,  but  it  must 
be  understood  that  the  word  can  be  used  only  with  refer- 
ence to  an  equation  of  one  unknown  number. 

The  student  must  not  confuse  the  word  root  as  here  used  with  the 
square  root  or  cube  root  or  some  other  root  of  expressions.  See 
Art.  151. 

233.  The  Solution  of  an  equation  is  the  process  by 
which  the  roots  are  found. 

EXERCISE  lOG. 

Pure  Quadratic  Equations. 

234.  If  x^=4  we  know  that  x  must  be  some  number 
which  raised  to  the  second  power  will  give  4.  Now, 
there  are  two  such  numbers,  -f  2  and  —2.  Therefore, 
.r=2  or  — 2.  Either  of  these  numbers  will  satisfy  the 
given  equation,  /.  e.,  will  render  the  given  equation  true. 
Also,  if  x'^  =  d  we  know  that  .r  is  a  number  which  raised 
to  the  second  power  will  give  9.  Either  +3  or  —3  will 
satisfy  the  given  equation.  Therefore,  x=S  or  —3.  So 
whatever  number  is  placed  equal  to  x'^  there  are  two 
numbers  of  opposite  signs,  but  otherwise  alike,  which 
will  satisfy  the  equation,  or  in  other  words,  there  are  two 
values  of  .^•  of  opposite  signs  but  otherwise  just  alike. 

235.  To  solve  a  pure  quadratic  equation  we  reduce  it 
so  that  all  the  unknown  terms  are  on  one  side  of  the 
equation  and  all  the  known  terms  on  the  other  side,  then 


PURE  QUADRATIC  EQUATIONS.        253 

we  divide  by  the  coefficient  of  the  square  of  the  unknown 
quantity,  and  lastly  extract  the  square  root  of  each  side 
of  the  equation. 

236.  In  solving  a  pure  quadratic  equation  we  usually 
write  both  values  of  the  unknown  quantity  at  once,  as  in 
the  equation  ;r^=4  after  extracting  the  square  root  of 
each  side  we  would  write  ;tr=±2,  using  the  double  sign 
db  to  show  that  either  -f  2  or  —2  will  satisfy  the  given 
equation  x^=4. 

The  student  may  think  that  we  should  write  the  double  sign  on 
doi/i  sides  of  the  equation  instead  of  on  one  side  only,  thus,  ±jc=  ±2', 
but  this  would  evidently  mean  x— 2  or  x=—2  or  —x=2  or  —x=  —  2. 
The  third  of  these  equations  is  really  the  same  as  the  second  and  the 
fourth  is  really  the  same  as  the  first,  so  that  we  really  get  no  more 
values  by  writing  ±x=  ±2  than  we  do  by  writing  x=  ±2. 

237.  Whenever  we  extract  the  square  root  of  each 
side  of  an  equation  we  should  write  the  double  sign  =b 
on  one  side  of  the  equation  obtained. 

Examples. 

Solve  the  following  equations : 

1.  x^  +  S=4. 

2.  jir2+3=7. 

3.  (;c24-l)-f-(^2+2)  +  (^2^3)=30G. 

4.  (>2-4)-f(j>;-+2)=^2_|_i4^ 

5.  2(ji:2  +  l)  +  3(x2  +  J)  =  5.i-. 

6.  3(x2-l)  +  4Gr2-2)=o;tr2-f39. 

7.  3j«;''-4=28+^^ 

8.  5x2-7=293  +  2jt:2. 

g.  (.r2-l)-Cr2-2)-Cr2-3)=54-3;»:^ 


2  54  QUADRATIC   EQUATIONS. 

10.   ^—  +  —77r--  +  —r, — =25-x^. 

O  10  o 

2.4,  ;»:     4 

"•  i^l  +  -5=^-  '3.  4=-- 

15.  12.v2-75=0. 

iG.  2Cr2-l)-3(;r2  +  l)+9=0. 

x^-\-l     x'^—1     .^     o. 
^7-  -2 8 ^°=°- 

XX  x 

19.  (2;r+l)2=4xH-2. 

20.  3;»:2-199=(;r+l)2-2.r. 

238.  The  equation  .r2=4  may  be  written  in  tlie  form 
;»;2_zj.— 0.  Now,  as  the  first  member  of  this  equation  is 
the  difference  of  two  squares,  it  may  be  factored,  and 
hence  the  equation  may  be  written  (.r— 2)(;f-f  2)=0. 

239.  The  product  of  two  or  more  factors  is  equal  to 
zero  whenever  any  one  of  the  factors  is  zero.  Therefore 
the  equation  (x— 2)(;t;+2)  =  0  can  be  satisfied  in  either 
of  two  ways:  first,  when  ;t'— 2=0,  i.  e.^  when  :r=2,  and 
second,  when  .r  +  2=0,  i.  e.,  when  .r=-— 2. 

As  another  example  take  the  equation 

5x2-9=2a-2  4-18. 

Transpose  all  the  terms  to  the  first  member  and  we  g2t 

5.r2^2.r2-9-18=0, 

or  3a'2-27=0. 

Dividing  by  3,  .;r2-9-=0. 

Factoring,  (jr— 3)(a'+3)=i0. 


AFFECTED   QUADRATICS.  255 

This  equation  can  be  satisfied  in  either  of  two  ways  : 
first,  when  x—S=0,  i.  e.,  when  jr=o,  and  second,  when 
^+3=0,  /.  e.,  when  :r=— 3. 

240.  By  the  method  illustrated  in  these  two  examples 
we  get  the  same  roots  as  would  be  obtained  by  the  former 
method.  Thus  we  have  another  method  of  solving  a 
pure  quadratic  equation,  viz.:  Collect  together  into  one 
term  all  the  unknown  numbers  and  into  another  term  all 
the  known  numbers ;  write  these  two  terms  on  the  same 
side  of  the  eq2iatio)i,  maki^ig  the  other  side  zero;  divide  both 
fnembers  by  the  coefficient  of  the  square  of  the  unknown 
number  (remembering  that  when  zero  is  divided  by  any- 
thifig  the  quotient  is  still  zero);  factor  the  7'esulling  first 
member;  put  each  factor  separately  equal  to  zero^  and  solve 
the  resulting  simple  equations. 

Examples. 

Solve  the  following  equations  by  the  method  just 
explained  : 
.21.  .;c2-100=0.  25.  (jir+2)2=4(;»r+5). 

22.  4.r-^--100=0.  26.  (.;ir+2)(.r+3)=5.r+42. 

23.  5.^2=80.  27.  .r2+;ir+l=jr4-101. 

24.  jr+-=— .  28.  .;t:2-2.r-3=33-2.;ir. 

'  X      X 

29.   (;e-+«  +  Z^)2  =  2(«4-^>r+2(«2^^2)^ 
30..(2.r+l)2=4a;-}-82. 

EXERCISE  107. 

Affected  Quadratics. 

241.  If  we  have  given  the  equation  .^^=25  we  solve 
it  by  the  preceding  exercise,  aad  find  x—±ih.  Similarly, 
if  we  have  the  equation  (j;-l- 1)2=25,  we  find  ;»:+l=±5. 


256  QUADRATIC    EQUATIONS. 

If  we  take  the  upper  s'gn  we  get  x+l=^B,  or  x—i,  and 
if  we  take  the  lower  sign  we  get  x-\-l  =  —o,  orjr=— 6. 
Thus  we  find  the  roots  of  the  equation  (x-\-iy=25y  or 
what  is  the  same,  ;tr2  +  2;f+l  =  25,  or  ^2 +  2;*:=  24. 

242.  Therefore,  to  solve  x'^+2x=24:,  we  first  add  1  to 
each  member  to  make  the  first  member  a  perfect  square, 
and  get  ^2  +  2;t-+l=25  ;  then  we  take  the  square  root  of 
each  member,  and  get  ;r+l  =  d=5,  whence  a'=4  or  —6. 

243.  Similarly,  to  solve  x'^-\-Gx=7y  we  add  to  each 
member  such  a  number  as  will  make  the  first  mem- 
ber a  perfect  square.  Plainly,  9  is  such  a  number. 
Therefore  jr^-f  6;r+9=16.  Next,  take  the  square  root 
of  each  member,  and  get  ;i;+3=±4,  whence  x—1  or  —7. 

244.  Similarly,  to  solve  afty  affected  quadratic  equa- 
tion, we  first  reduce  the  equation  to  a  form  where  the 
terms  containing  x^  and  x  are  in  the  first  member  and 
the  term  not  containing  x  is  in  the  second  member; 
second,  if  the  coefficient  of  x^  is  not  unity,  divide  each 
member  of  the  equation  by  that  coefficient,  so  that  the 
coefficient  of  :i:^  shall  be  unity;  third,  add  to  each  member 
of  the  equation  such  a  number  as  will  make  the  first  mem- 
ber a  perfect  square,  and  then  take  the  square  root  of  each 
member  and  solve  the  resulting  simple  equations. 

245.  Adding  to  a  given  expression  a  number  that  will 
make  the  sum  a  perfect  square  is  called  Completing  the 
Square. 

24G.  When  the  coefficient  of  x"^  is  unity  v/hat  number 
is  it  that  we  must  add  to  each  member  of  an  equation  to 
make  the  first  member  a  perft^t  square  ?  To  answer  this 
let  us  see  how  a  perfect  square  is  produced.  We  know  that 


AFFECTED   QUADRATICS.  257 

and  (x—a)'^=x^—2ax-\-a". 

Notice  here  that  whatever  luiinber  is  represented  by  a 
the  third  term  is  the  square  of  one-half  the  coefficient 
of  Ji' :  hence  the  number  to  be  added  to  each  member  o: 
the  given  equation  is  the  square  of  one-half  the  coefficient 
of  X. 

247.  Hence,  to  solve  any  affected  quadratic  equation: 
/.  Reduce  to  a  form  wJiere  boih  x'^  and  x  are  in  the  first 

member  and  all  terms  not  containing  x  are  in  the  second 

member. 

II.  If  the  coefficient  of  x^  is  not  already  nnity,  divide 
each  member  of  the  equation  by  that  coefficient,  thus  making 
the  coefficient  of  x-  unify. 

III.  Com  pie  fe  the  square  by  adding  to  each  member  the 
square  of  one- half  the  coefficient  of  x. 

IV.  Extract  the  square  root  of  each  member  of  the  equa- 
tion and  solve  the  resulting  simple  equations. 

Examples. 

Solve  the  following  equations  : 

1.  jr2-f4.r=5.  ii.  x--^^x=-\o. 

2.  j»r2-|-G.r=lG.  12.  3a-2H-12x=-3G. 

3.  2A-2  +  20jr=43.  13.  2.r2-f  10ji'=100. 

4.  x'^  +  ox=l^.  14.  x^—ax^Q^a"^. 

5.  x''-^bx=Z(j.  15.  a"2-2^A^=8«^ 

G.  3.r'H-C-r=9.  16.  3A-2-12^.r==63«^ 

7.  4;»;2-4ji'=8.  17.  4;i'2--12^x=16a2. 

8.  x'^-lx=-(j.  18.  5;r2-25A'=-20. 

9.  ^2_10ji:=— 0.  ig.  x'^—x=2. 

10.  2j;-  — 15;t=50.  20.  x'^+x=a''--\-a. 

17 


258  QUADRATIC    EQUATIONS. 

248.  The  method  already  given  will  enable  us  to 
solve  any  affected  quadratic  equation  that  may  be  given, 
but  frequently  it  willoblige  us  to  use  fractions,  and 
imless  the  terms  of  the  fractions  are  small  numbers  it 
will  be  easier  to  complete  the  square  by  another  method, 
which  we  will  now  consider. 

249.  We  know  that 

and  {ax—b')-=a'^x''—2abx-\-b'^y 

so  that  each  of  these  two  second  members  is  a  perfect 
square.  We  therefore  seek  to  reduce  the  given  equation 
so  that  the  first  member  shall  be  in  the  form  of  one  oi 
these  two  second  members. 

250.  Notice  two  things :  first,  that  the  coefficient  of 
x"^  is  a  perfect  square,  and  second,  that  the  third  term 
equals  the  square  of  the  quotient  obtained  by  dividing 
the  second  term  by  tw'ce  the  square  root  of  the  first 
term.  Therefore,  to  reduce  the  first  member  of  any 
given  quadratic  equation  to  one  of  the  two  forms 

a'^x''-  +  2abx^b''- 
or  a''-x-—2abx-\-b''-, 

I.  Reduce  the  equation  so  ihat  the  terms  containing  x"^ 
and  X  shall  be  in  the  first  member  and  all  terms  not  contain- 
ing X  shall  be  in  the  second  member. 

II.  Mnliiply  each  member  of  the  equation  by  snch  a  mem- 
ber as  will  make  the  coefficient  0/  x-  some perfed  square. 

III.  Add  to  each  member  the  square  of  the  quotieni 
obtained  by  dividing  the  second  term  by  twice  the  square 
root  of  the  first  term. 

The  rest  of  the  process  of  solution  is  like  that  already 
given,  viz.:  take  the  square  7'oot  of  each  member  and  solve 
the  resulting  simple  equations. 


AFFECTED    QUADRATICS.  259 

Let  us  solve  by  this  method  the  equation 

Transpose  2x  and  11  and  we  get 
Sx-'-}-5x=22. 
Multiply  each  member  by  3  or  12  or  27  or  3  times  any 
square  number  and  the  coeffirient  of  x-  will  be  a  perfect 
square.     Take  the  first  of  these  multipliers  and  we  get 

9A-2-fl5j«r=6G. 
Add  to  each  member  (V")^'  or  (f)-,  and  we  get 

ar2  +  15;i;+-2/=66  +  -2/=lf«. 
Take  the  sqaare  root  of  each  member  and  we  get 

3a-4-|=±-V-. 
Hence,  dx=G  or  —11,  and  x=2  or  — -V'- 

If  we  had  multiplied  by  12  instead  of  3  we  would  have 
obtained  30x2  +  G0.r=2G4. 

Therefore,      3G;i-2+60j»;+ 25 =204  +  25= 239. 
Hence,  G.r-f-5=±17. 

Hence,         G.r=12  or  -22,  and  x=2  or  — U. 

Examples. 

Solve  the  following  equations  : 

21.  3ji:2+4x=7.  31.  3^-2— 24=G.r. 

22.  3jt:2-fG.r=24.  32.  2.r2-22x=— GO. 

23.  4j>;--5x=2G.  33.  2;i:2  +  10;i;=300. 

24.  9.r2+G.v-48=0.  34.  3.r2-10ji;=200. 

25.  lSx^-—Sx—m=0.  35.  2;»;2_3^-^io4. 

26.  5jt:--7;r=24.  36.  3.r2+7jt:-370=0. 

27.  2.r2-35=3.r.  37.  4x^^7x-{-^=0. 
23.  3;»;2_50=5jtr.  38.  5x''-},x—;'^=0. 

29.  i;r2-3.r+ij=0.  39-  |-^;-t-r=— J?. 

x"^     Sx  ,  ^     f.  x"^     X     I     ^ 

3°'  T-T+^='^-        40.  -ir-2+G=°- 


260  QUADRATIC  EQUATIONS. 

251.  Literal  quadratic  pq nations  may  be  solved  the 
same  as  numerical  ones.  But,  after  the  square  is  com- 
pleted, the  second  member  will  be  a  literal  instead  of  a 
numerical  expression,  and  hence  the  square  root  usually 
cannot  be  taken,  and  so  will  have  to  be  indicated.  Thus, 
to  solve  x'^-j-4ax+d=0,  we  proceed  as  follows  : 

Transpose  d,  x'^-\-4:ax=—b. 

Add  Aa-  to  each  member. 

Take  the  square  root  of  each  member, 

x-]-2a=-^VAa'-  —  b. 
Transpose  2a  x=  —  2<2d=  V \a "-  —  b. 

One  value  o{ x  is  —2a-\-V Aa'^  —  b  and  the  other  value  is 

Examples. 
Solve  the  following  equations  : 

41.  x''--\-2ax=b.  46.   2x''-—Q>ax—Ab=0. 

42.  x'^-\-Aax=b.  47.  bx'^—lax-\-b=^0. 

43.  2jt-2-f3^.r=4(^.  48.   ax'^-{-bx+c=0. 

44.  x'^—bax=1b.  49.  ax'^  —  bx=c. 

45.  x'^  —  (dax—ob=0.  50.  ax"^ -\- a"^ x= a^ . 

252.  When  all  the  terms  of  an  affected  quadratic 
equation  are  transposed  to  the  left  member,  making  the 
right  member  zero,  the  equation  may  be  solved  by  fac- 
toring if  the  resulting  first  member  can  readily  be  ex- 
pressed as  the  product  of  two  factors  each  of  the  first 
degree  with  respect  to  the  unknown  numbers.  Thus,  to 
solve  the  eqr»ition  x'^  —  bx=—Q,  we  transpose  —6,  and 
obtain  x'^—bx+Q)=0.  By  the  method  of  exercise  88  we 
find  the  factors  of  the  left  member  to  be  x—2  and  x—S. 
Therefore  the  equation  may  be  written  (x— 2)(a-— 3)=0, 
which  is  satisfied  if  ;»;=2  or  if  ;»r=3. 


PROBLEMS.  261 

253.  This  method  of  solving  an  affected  quadratic 
cannot  be  used  to  advantage  unless  the  left  member  is 
easily  factored,  but  when  easily  factored  this  is  probably 
the  easiest  way  to  solve  them. 

Examples. 

Solve  by  factoring  the  following  equations  . 

51.  x-'-i-dx+Q^O.  56.  x''-5x=U. 

52.  x--\-llx=—^0.  57.  jr2+j»;=80. 

53.  x''—x=Q.  58.  A'2+jr=12. 

54.  .r2  +  7;i:=— 12.  59.  :r--7:r+12=0. 

55.  x2-lljt;+30=0.  Go.  2.r2-fGjtr=,i;2--3.r— 14. 

61.  2x--\-Sx-{-4=x'^—Sx—l. 
G2.  x2  +  2jr+l  =  6.r-fG. 
63.   3;i:2  +  12;i:-10=2;t'2^2,r-31. 
G4.  x2  =  G.r-rx  70.  2x--40ji-=4ji--240. 

65.  ;r2==-4.r  +  21.  71.   x^--<ia-\-l?')x  +  ad=0. 

66.  ji'2  =  — 4;»;+5.  72.  x'^  —  (a+l)x-^a=0. 

67.  ;r2— 49=10(;»:— 7).         73-  A'-  +  («  +  ^Xr+^^=0. 

68.  2jf2+60.r=-400.  74.  x2^(a  +  l)jr  +  a=0. 
eg.   10ji;2  4-G00a-=-8000.    75.  .r2  4-(r.3+l).v+rtZ'=0. 

EXERCISE  103. 

Prodlems  Leading  to  Quadratic  Equations. 

251.  To  solve  a  problem  the  first  thing  to  do  is  to 
form  an  equation  the  result  of  solving  which  will  fulfill 
all  the  requirements  of  the  problem.  The  formation  of 
this  equation  is  sometimes  a  great  difficulty  to  students, 
but  after  the  equation  is  once  written    down    there    is 


262  QUADRATIC    EQUATIONS. 

usually  little  or  no  difficulty  with  a  problem.  The  diffi- 
culty here  spoken  of  may  b^  largely  overcome  if  the 
student  will  keep  in  mind  the  fact  that  the  equation  is 
formed  by  using  the  unknown  number  in  exactly  the 
same  way  we  would  uss  any  assumed  result  to  see 
whether  or  not  thi^  assumed  result  were  right.  This  is 
illustrated  in  the  following  problem. 

I.  A  train  travels  GOO  miles  at  a  uniform  rate  of  speed. 
If  the  rate  had  been  10  miles  more  an  hour  the  journey 
would  have  taken  5  hours  less.  Find  the  rate  of  the 
train. 

Let  us  see  if  40  miles  an  hour  is  the  rate.  If  the  train  goes  40 
miles  an  hour,  to  go  GOO  miles  will  require  -^j  or  1 5  hours.  And  if 
the  rate  were  10  miles  more  an  hour  it  would  be  50  miles  an  hour, 
and  to  travel  600  miles  at  this  rate  would  require  %V  or  12  hours. 
By  the  first  supposition  (40  miles  an  hour)  it  requires  15  hours, 
and  by  the  second  supposition  (50  miles  an  hour)  it  requires  12  hours. 
As  this  last  result  (12  hours)  is  not  5  hours  less  than  the  first  result 
(15  hours)  we  conclude  that  the  rate  is  7iot  40  miles  an  hour. 

Now  let  us  see  if  30  miles  an  hour  is  the  rate.  To  travel  GOO  miles 
at  30  miles  an  hour  requires  \"(f-  or  20  hours,  and  to  travel  600  miles 
at  40  miles  an  hour  requires  -Y/-  or  15  hours  ;  and  as  20  —  15=5,  (/.  e. 
the  second  time  is  5  hours  less  than  the  first,)  we  conclude  that  the 
rate  is  30  miles  an  hour. 

Now  to  form  an  equation  we  say,  let  x  represent  the  number  of 

miles  an  hour  the  train  travels  ;  then  to  travel  600  miles  at  x  miles  an 

fiOO  , 
hour  requires hours,  and  to  travel  GOO  miles  at  jc-i-10  miles  an 

hour  requires —  hours,  and  as  the  tim2  in  the  second  instaacs  is 

^  1-  J «) 

6  hours  less  than  in  the  first  instance,  we  have 

GOO        GOO  _ 

X       X  \-\\f~ 

From  this  equation  we  have 

5.r- -t-50.r=600'ar-MO)  — 600;c, 

or  5a;- -|-50a;=:G000, 

or  a;3  4- 10^=1200. 


PROBLEMS.  263 

Completing  the  square,  we  have 

x2-|-l0r+25=1225. 
Extracting  the  square  root,  we  have 

Hence  j:=30  or  —  40. 

The  first  of  these  results,  30,  agrees  with  the  conclusion  reached 
above,  but  here  another  question  arises, — what  is  to  bs  done  with 
the  result  —40  ?  So  far  as  the  algebraic  work  goes,  —40  is  as  good  a 
result  as  30,  but  to  speak  of  a  train  traveling  —40  miles  an  hour  is 
something  void  of  meaning,  so  this  result  is  rejected  and  30  is  re- 
tained as  the  true  result. 

255.  It  will  often  happen,  as  in  the  example  just 
worked,  that  the  solution  or  the  equation  formed  as 
already  described  leads  to  a  result  which  does  not  apply 
to  the  problem  we  are  solving.  The  reason  of  this  is  that 
the  algebraic  statemt^nt  of  the  prrblem  (by  means  of  the 
equation  formed  as  above  described)  is  more  general  than 
the  statement  in  words.  It  will,  however,  usually  be 
quite  easy  to  select  which  result  belongs  to  the  problem 
we  are  solving,  and  then  we  can  reject  the  other  result 
as  inapplicable. 

2.  A  c'.stern  can  be  filled  by  two  pipes  in  33^  minutes. 
If  the  smaller  p'pe  takes  15  minutes  more  than  the  larger 
one  to  fill  the  cistern,  in  what  time  will  it  be  filled  by 
each  pipe  singly  ? 

Let  us  see  if  the  smaller  pipe  would  fill  the  cistern  in  43  minutes. 
If  so,  the  larger  pipe  would  fill  the  cistern  in  ."0  minutes.  If  larger 
pipe  will  fill  the  cistern  in  30  minutes  it  would  fill  g^j  of  the  cistern  in 
1  minute,  and  if  smaller  pipe  will  fill  the  cistern  in  45  minutes  it 
would  fill  5^5  of  the  cistern  in  1  minute.  Therefore,  together  they 
would  fill  a^y  +  ^j  =  iV  ^^  cistern  in  1  minute.     But  together  they  fill 

— -=-^  of  cistern  in  I  minute,  aaJ  as  ^ij  is  not  equal  to  ^ gjy  we  con- 
clude that  45  minutes  is  nai  the  time  the  smaller  pipe  would  fill  the 
cistern. 


264  QUADRATIC    EQUATIONS. 

Let     r=number  of  minutes  required  to  fill  cistern  by  smaller  pipe, 
then  ^-15  =  number  of  minutes  required  to  fill  cistern  by  larger  pipe. 
If  larger  pipe  will  fill  the  cistern  in  x—}3  minutes  it  would  fill 

of  cistern  in  1  minute,  and  if  smaller  pipe  would  fill  cistern  in 


X  minutes  it  would  fill  -  of  cistern  in  1  minute.     Therefore  the  two 

X 

pipes  together  would  fill rr  +  -  or  -; ~  of  cistern  in  1  minute. 

^^         °  x—\o     X       jr(x— lo) 

But  together  they  fill  — —  or  ■:—-  of  cistern  in  1  minute.    Therefore 

OO^r  1  \j\J 

a         2r-15 


lUJ     a  (a:— 15) 
Clearing  of  fractions,  we  get 

ax-  -4:)a;=2()0.r- 1500. 
Transposing  200x,  3x--  215.t;=  -  loOO. 

Multiply  by  12,  3Gr3  —  29-IOx=  — 18000. 

Add  (21:5)^  to  each  member  to  complete  the  square, 

36x-  — 2!)40.'c  -1-  (10025  =  42023. 
Extract  the  square  root  of  each  member, 

Ga:-24.*=±205. 
Hence,  6-r=450  or  40. 

Therefore,  .r=75or6j. 

From  this  it  appears  that  the  smaller  pipe  would  fill  the  cistern  in 
either  75  or  6^  minutes,  but  evidently  (5 ^  is  not  admissible,  for  it  takes 
the  smaller  pipe  15  minutes  7nore  to  fill  the  cistern  than  it  takes  the 
larger  pipe  some  time  to  fill  the  cistern.  So  it  is  plain  that  it  must 
take  the  smaller  pipe  more  than  15  minutes  to  fill  the  cistern.  We 
therefore  reject  the  result  G}  as  being  inadmissible;  the  other  result, 
75,  is  admissible,  however,  and  satisfies  the  requirements  of  the 
problem.  Hence,  it  takes  the  smaller  pipe  75  minutes,  and  therefore 
the  larger  one  GO  minutes  to  fill  the  cistern  alone. 

3.  A  merchant  sold  some  damaged  goods  for  $72, 
v.'h:cli  was  a  lo.S3  per  cent,  of  \  of  the  number  of  dollars 
the  goods  co3t.     Find  the  cost  of  the  goods. 

Let  ^        a;=number  of  dollars  goods  cost, 

then  -=rloss  per  cent. 

Therefore,  .^-entire  los? 

ouO 


PROBLEMS.  265 

By  the  statement  in  the  problem  we  have 

^2  +  21600  =  300^:. 
a;^-300j;=-21GOO, 
a;'— 300ar  +  22,>(i0=900, 
ar— lo0=:±30, 
rc=lS0  or  120. 
Each  of  these  answers  fulfills  all  the  requirements  of  the  problem, 
and  each  is  admissible. 

4.  One  of  two  numbers  is  f  the  other  one  and  the  sum 
of  their  squares  is  20S.     Find  the  two  numbers. 

5.  The  product  of  two  numbers  is  750  and  the  quotient 
of  the  greater  divided  by  the  less  is  3^.  Find  the  numbers. 

6.  Divide  the  number  103  into  two  parts  whose  prod- 
uct is  2400. 

7.  Divide  the  number  GO  into  two  such  parts  that  the 
quotient  of  the  greater  divided  by  the  less  may  equal  1 
more  than  twice  the  less. 

8.  A  merchant  bought  a  quantity  of  cloth  for  S120  ;  if 
he  had  bought  6  yards  more  for  the  same  sum  the  price 
per  yard  would  have  been  $1  less.  How  many  yards  did 
he  buy,  and  what  was  the  price  per  yard  ? 

g.  A  merchant  sold  some  goods  for  $39,  and  in  so  doing 
gained  as  much  per  cent,  a-,  the  goods  cost  him.  What 
was  the  cost  of  the  goods  ? 

10.  Find  a  number  such  that  3  more  than  twice  the 
number  multiplied  by  3  less  than  twice  the  number  may 
give  a  product  of  112. 

11.  A  man  traveled  105  miles,  and  then  found  if  he 
had  gone  2  miles  less  per  hour  he  would  have  been  6 
hours  longer  on  his  journey.  How  many  miles  did  he 
travel  per  hour  ? 


266  QUADRATIC    EQUATIONS. 

12.  A  man  bought  two  farms  for  12800  each  ;  the 
larger  contained  10  acres  more  than  the  smaller,  but  he 
paid  $5  more  per  acre  for  the  smaller  than  for  the  larger. 
How  many  acres  were  there  in  each  farm  ? 

13.  A  merchant  sold  two  pieces  of  cloth  which  together 
contained  40  yards,  and  received  for  each  piece  twice  as 
many  cents  per  yard  as  there  were  yards  in  the  piece. 
For  the  smaller  piece  he  received  2^-  as  much  as  for  the 
larger  one.     How  many  yards  were  there  in  each  piece  ? 

14.  The  distance  around  a  rectangle  is  200  feet,  and 
the  area:  is  1344  square  feet.  Find  the  length  and  breadth 
of  the  rectangle. 

15.  The  length  of  a  rectangle  is  10  feet  more  than  the 
breadth,  and  the  area  is  GOO  square  feet.  Find  the  length 
and  breadth  of  the  rectangle. 

16.  There  are  three  lines,  the  first  two  of  which  are 
f  of  the  third,  and  the  sum  of  the  squares  described  on 
these  three  lines  is  33  square  feet.  Find  the  lengths  of 
these  lines. 

17.  A  flower  bed  9  feet  long  and  6  feet  wide  has  a  path 
around  it  whose  area  is  equal  to  the  area  of  the  bed  itself. 
What  is  the  width  of  the  path  ? 

18.  A  number  consists  of  two  digits  one  of  which  is 
the  square  of  the  other,  and  if  54  be  added  to  the  number 
the  digits  are  reversed  in  order.     What  is  the  number? 

19.  Find  a  number  such  that  if  it  be  added  to  94  and 
again  subtracted  from  94  the  product  of  the  sum  and 
dijfference  thus  obtained  shall  be  8512. 

20.  A  man  bought  a  number  of  horses  for  $10000  ; 
each  cost  4  times  as  manj^  dollars  as  there  were  horses. 
How  many  horses  did  he  buy  ? 


PROBLEMS.  267 

21.  Find  a  number  whose  square  is  greater  than  the 
number  itself  by  306. 

22.  Find  a  number  such  that  if  its  third  part  be  multi- 
phed  by  its  fourth  part  and  to  the  product  5  times  the 
number  be  added  the  sum  exceeds  200  by  as  much  as 
the  number  required  is  less  than  280. 

23.  A  man  bought  a  horse  and  sold  it  again  for  $119, 
by  which  means  he  gained  as  many  per  cent,  as  the  horse 
cost  him  dollars.  How  many  dollars  did  the  horse  cost 
him  ? 

24.  Find  the  fortunes  of  three  persons,  A,  B,  and  C, 
from  the  following  data  :  For  every  $5  which  A  has 
B  has  $9  and  C  has  $10  ;  mo'eover,  if  we  multiply  A's 
money  by  B's,  and  B's  money  by  C's,  and  add  both 
products  to  the  united  fortunes  of  all  three  we  shall  get 
$8832.     How  much  money  has  each  ? 

25.  The  combined  area  of  two  squares  is  9G2  square 
feet,  and  a  side  of  one  square  is  18  feet  longer  than  a 
side  of  the  other.     What  is  the  size  of  each  square  ? 

26.  A  square  field  conta'ns  a  number  of  square  rods 
equal  to  2G0  more  than  32  times  its  perimeter.  How 
many  rods  in  one  side  of  the  square  ? 

27.  F!nd  two  numbers  whose  sum  is  21  and  the  sum 
of  whose  squares  is  225. 

28.  Find  two  numbers  whose  product  is  480  and  the 
difference  of  whose  squares  is  3536. 

29.  What  is  the  price  of  oranges  when  10  more  for 
$1.20  lowers  the  price  1  cent  each  ? 

30.  The  sum  of  tht*  ages  of  a  father  and  son  is  80  years, 
and  -J  of  the  product  of  their  ages  in  years  exceeds  5  times 
the  father's  age  by  200  years.     What  is  the  age  of  each  ? 


268  QUADRATIC   EQUATIONS. 

31.  A  certain  number  is  the  product  of  three  consecu- 
tive whole  numbers,  and  if  it  is  divided  by  each  one  of 
these  three  factors  in  turn  the  sum  of  the  three  quotients 
thus  obtained  is  7G7.     What  is  the  number? 

32.  The  sum  of  the  squares  of  three  consecutive  odd 
numbers  is  83.     What  are  the  numbers  ? 

33.  The  sum  of  the  squares  of  four  consecutive  even 
numbers  is  120.     What  are  the  numbers? 

34.  Divide  the  number  18  into  two  such  parts  that 
their  product  shall  exceed  30  times  their  difference  by  20. 

35.  In  a  bag  which  contains  coins  of  silver  and  gold 
each  silver  coin  is  worth  as  many  cents  as  there  are  gold 
coins,  and  each  gold  coin  is  worth  as  many  dollars  as 
there  are  silver  coins,  and  the  whole  is  worth  $525.  How 
many  gold  and  how  many  silver  coins  in  the  bag? 

36.  A  room  whose  length  exceeds  its  breadth  by  8  feet 
is  covered  with  matting  4  feet  wide,  and  the  number  of 
yards  in  length  of  the  matting  exceeds  f  the  number  of 
feet  in  breadth  of  the  room  by  20.  Find  the  length  and 
breadth  of  the  room. 

37.  There  are  two  numbers  whose  difference  is  7,  and 
half  their  product,  plus  30,  is  equal  to  the  square  of  the 
smaller  number.     What  are  the  numbers  ? 

38.  A  and  B  start  together  on  a  journey  of  36  miles. 
A  travels  1  mile  per  hour  faster  than  B  and  arrives  3 
hours  before  him.     Find  the  rate  of  each. 

39.  Two  workmen,  A  and  B,  are  engaged  to  work  at 
different  wages.  A  works  a  certain  number  of  days  and 
receives  $27,  and  B,  who  works  1  day  less  than  A,  re- 
ceives $34.  If  A  had  worked  2  days  more  and  B  2  days 
less,  they  would  have  received  equal  amounts.  Find  the 
number  oi  days  each  worked. 


EQUATIONS    SOLVED    LIKE   QUADRATICS.        269 

EXERCISE  109. 

Equations  Solved  Like  Quadratics. 

256.  It  is  bej'ond  the  scope  of  this  book  to  treat  equa- 
tions of  a  higher  degree  than  the  second  expressed  in 
general  form,  but  there  are  a  few  equations  of  higher 
degree  which  may  be  solved  by  the  methods  of  this 
chapter. 

257.  We  have  had  such  equations  as  x^  — lo.i-f  3G=0, 
and  have  seen  that  such  equat'ons  are  easily  solved.  Now 
it  is  plain  that  we  couhi  use  some  other  symbol  in  place 
of  X  to  designate  an  unknown  number.  Thus  we  might 
have  an  equation  where  r"'  stands  in  place  of  Jtr,  and  then 
of  course  jK"'  would  stand  in  place  of  jt-,  and  the  equation 
would  be 

jj/^-13)/2-f3G=0, 
from  which,  by  solving  in  the  usual  way,  regarding  jv^ 
temporarily  as  the  unknown  number,  we  obtain 

j,2=4or9. 
Hence  jj/=±2  or  ±3. 

In  a  similar  manner  we  could  treat  equations  in  v/hich 
more  complex  expressions  stand  in  place  of  x'^-  and  x  in 
the  equations  before  used,  but  whatever  expression  stands 
in  place  of  x  the  square  of  that  expression  must  stand  in 
place  of  x'^ ,  else  the  equation  cannot  be  solved  by  the 
methods  of  this  chapter. 

Examples. 
Solve  the  following  equations  : 

1.  .r4-29.r2-f  100=0.  3.  y-17y^  +  lG=0. 

2.  .r«-35a-5 +  210=0.  4.  jj/+8l/J'+15=0. 

5.  (.r-f3)6-28(;«r-f3)«  +  27=0. 


:270  QUADRATIC    EQUATIONS. 

6.   ^2+_L_^2  +  i_. 

-  ('+l)*-¥(-l)+'=»- 

10.   (;r2-5.r+G)'-^  =  14(;t'2_5.^^G)_24=0. 

EXERCISE  110. 

Theory  of  Quadratic  Equations. 

253.  The  methods  a'ready  explained  will  enable  us 
to  so  ve  a  quadratic  equation  in  which  letters  are  used 
to  stand  for  the  known  numbers  in  the  equation,  but  01 
course  we  nuiit  not  use  the  same  letter  to  stand  for  a 
known  number  that  we  use  to  stand  for  an  unknown 
number. 

Let  us  solve  the  equation  x"-  -\-ax-\-b=^^. 

Transposed,  x'^-\-ax=  —  b. 

•Complete  square, 

a  ,  a"-      a'^  a-— 41? 

x-^-hax-^--=--—d=--~- 

Extract  square  root,  .r+o==i=A/  -^ 

Hence,  x=  - 1±  ^^-"Z-l.. 

From  this  we  see  that  one  root  ol  the  given  equation 
a  ,      (a^-Tb 
=  ~2  +  A/""4""' 
.and  the  other  root  of  the  given  equation 
a         la-—4d 


THEORY  OF  QUADRATIC  EQUATIONS.     2/1 

259.  It  thus  appears  that  there  are  two  roots  to  a  qiiad- 
rat'c  equation,  but  for  certain  values  of  a  and  b  these  two 
roots  are  exactly  the  same.  This  will  be  the  case  when 
«2_4^=0,  for  then  the  number  under  the  radical  sign 
reduces  to  zero,  and  each  root  of  the  equation  reduces  to 

—-5.     In  this  case  there  is  in  reality  only  one  value  of  ;r 

that  will  satisfy  the  equation.  Still,  instead  of  saying 
that  there  is  only  one  root  of  the  equation,  we  say  that 
there  are  two  roots  but  that  these  two  are  cqtial  to  each 
other.  Of  course  this  is  only  another  way  ot  saying  that 
there  is  only  one  root,  but  the  advantage  of  this  mode  of 
expression  will  be  apparent  as  we  proceed. 

260.  Let  us  now  find  the  sum  of  the  roots  of  the 
equation  x''--\-ax-^b=^.  


One  root=-|+y^-j-^ 
a         \~^- 


other  root= 


sum=— flj 

In  the  quadratic  equation  ;r-+a.r+^=0  the  coelTicient 
oi  x"-  is  unity,  but  a  and  b  stand  for  any  numbers.  Hence 
in  any  quadratic  equation  where  all  the  significant  terms 
are  in  the  left  member  and  the  right  member  is  0,  and 
where  the  coefficient  of  x^  is  unity,  the  sum  of  the  roots  is 
equal  to  the  coeffident  of  x  with  its  sign  changed. 

261.  Let  us  now  find  the  product  of  the  roots  of  the 
equation  x"^ -\- ax -\- b ^  ^ . 

The  product  required  may  be  expressed  thus : 
/     a         \a'—\b\(     a         \a'^—\b\ 

\r^i'\-^r)\-T-\—A-) 


2y2  QUADRATIC   EQUATIONS. 

Notice  that  this  is  the  product  of  the  sum  and  difierence 
of  two  numbers,  which,  as  we  have  already  learned,  is 
equal  to  the  difference  of  their  squares.  Therefore  the 
product  of  the  two  roots  equals 

a-      /a'-—4d\      , 

Hence  in  any  quadratic  equation  where  all  the  signifi- 
cant terms  are  in  the  left  member  and  the  right  member 
is  0,  and  where  the  coefficient  o(  x^  is  unity,  ^/le  pi'oditc, 
of  the  roots  is  equal  to  the  tcmi  not  containiiig  x. 

262.  The  last  two  articles  show  just  the  relation  be- 
tween the  known  numbers  in  an  equation  and  the  roots 
of  the  equation,  and  from  the  results  reached  we  are 
enabled  to  form  an  equation  which  shall  have  any  desired 
roots.  For  example,  if  we  wish  to  form  the  equation 
whose  roots  are  3  and  5,  we  know  that  the  coefficient  of 
.r^  will  be  1,  the  coefficient  of  x  with  its  sign  changed 
will  be  the  sum  of  the  roots,  /.  <?.,  in  this  ca.-:e  the  coeffi- 
cient of  x  will  be  —8,  and  the  term  not  containing  x  will 
be  the  product  of  the  roots,  i.  e.,  in  this  case  the  term 
not  containing  x  will  be  15.  Hence  the  equation  is 
.;t:'  — 8x-f  15=0. 

Examples. 

Form  the  equations  whose  roots  arc  the  following 
given  numbers : 

1.  2  and  3.  G.  4  and  G.  ii.  5  and -3. 

2.  3  and  G.  7.  3  and  —1.  12.   —7  and  — o. 

3.  1  and  7.  8.  2  and  2.  13.   12  and  —12. 

4.  G  and  — G.  9.   —4  and  —4.  14.  f  and  f . 

5.  3  and  3.  10.   10  and  20.  15.  2  and  0. 


THEORY  OF  QUADRATIC  EQUATIONS.     273 

263.  In  Art.  258  we  solved  the  equation  x'^-\-ax+0=0 
and  obtained  for  the  value  of  x 


a        ja- 


-Ab 


4 

This  result  ma}^  be  used  as  a  formula  for  finding  the 
roots  of  any  quadratic  equation  of  the  iox\\\  x''- -\-ax-\rb=^, 
i.  e.,  any  quadratic  equation  in  which  the  coefficient  oi  x'^ 
is  unity  and  all  the  terms  are  in  the  left  member.  To 
obtain  the  roots  of  any  quadratic  equation  of  this  form 
we  have  only  to  substitute  in  the  place  of  a  and  b  in  the 
result  given  above  their  numerical  va'ues.  Thus  we  may 
obtain  the  roots  of  a-- +  o.v— 70=0  by  writing  3  in  place 
of  a  and  —70  in  place  of  b.  Making  this  subjtitutiou,  we 
get  for  the  roots  


8_^    /9  +  -280 
=  -f±JJ:=7or-10. 


E.XAMPLES. 

In  this  manner  find  the  roots  of  the  following  equations: 

16.  .r2  +  12x-hll=0.  21.  A'2-f  lG.r+28=0. 

17.  .;c2-12.r+5|=0.  22.  A-2  +  3x-lJ=0. 

18.  a:2  +  15a'-|-31^=0.  23.  x''-2x-\-\^0. 

19.  .^2_|.Q^_433=0.  24.  .^•2-f7.r+G=0. 

20.  ;t2_20jr-44=0.  25.  .r--5.r-2J=0. 

264.  Let  us  try  to  use  the  same  method  to  find  the 
roots  of  the  equation  x'^-\-2x+S—0. 
By  substituting,  w**  have  for  the  roots 


-l±^i^=-l±^=7. 


13 


274  QUADRATIC    EQUATIONS. 

But  there  is  no  number  which  squared  will  give  —7,  and 
so  —7  has  no  square  root.  Nevertheless,  such  expres- 
sions as  l/  — 7  do  frequently  occur  in  Algebra,  and  are 
called  impossible  riumbers  or  imaginary  expressions. 

265.  An  Imaginary  Expression  is  any  expression 
which  contains  one  or  more  terms  in  which  there  is  an 
indicated  even  root  of  a  negative  number.  By  even  root 
we  mean  the  square  root,  fourth  root,  sixth  root,  etc.  Thus 

l/^,     4+T/-2,     a-^b-V~^, 
are  imaginary  expressions. 

266.  By  way  of  distinction  those  expressions  which 
are  not  imaginarj^  are  called  Real.  All  of  the  expres- 
sions with  which  we  have  had  to  deal  heretofore  have 
been  real. 

267.  By  inspecting  the  expressions  for  the  roots  of  the 
equation  jr^+^-r-f  <^=0  we  can  tell  when  the  two  roots 
are  real  and  when  imaginary,  for  the  roots  will  be  real 
when  the  expression  under  the  radical  sign  is  positive, 
and  imaginary  when  that  expression  is  negative.     The 

expression  under  the  radical  is  — j —  ^^^^  i^  this  expres- 
sion a"^  must  always  h^  positive  becaurc  it  is  the  square  of 

a^—\b 
some  number;  so  it  is  plain  that  the  expression  — - —  is 

positive  when  <2-  is  greater  than  4^,  and  negative  when 
a"^  is  less  than  \b.  Hence,  the  roots  are  real  when  «^  is 
greater  than  4^,  and  imaginary  when  a-  is  less  than  4^. 

If  the  quadratic  equation  that  is  given  is  in  a  different 
form,  we  can  still  find  when  the  roots  are  real  and  when 
imaginary  just  as  easily  as  was  done  in  the  case  just 
considered. 


THEORY  OF  QUADRATIC  EQUATIONS.     27$ 

Take,  for  example,  the  equation  ax'^  +  dx+c^O, 
Solving  this,  we  get 

and  here  the  express'on  under  the  radical  sign  is  positive 
when  d-  is  greater  than  4ac,  and  negative  when  d"^  is  less 
than  Aac.  Therefore,  the  rooti  are  real  when  d"^  is 
greater  than  4ac,  and  imaginary  when  ^-  is  less  than  4ac. 

Examples. 

Solve  the  following  equations  and  determine  when  the 
roots  are  real  and  when  imaginary: 

26.  x'^—2ax+2d=0.  31.  ax^-—4dx—4=0. 

27.  ax'^-2dx-\-Sc=-0.  32.  x''-4ax-\-5=0. 

28    x''-2-x-~=0.  33.  x''-4x+a==0. 

a      .  a 

29.  2x'^-\-Zax=^hb.  34.  a'x''--\-4bx-4c^^. 

30.  ax'^-\-Ux-\-^ab^^.  35.  2ax''^-Zbx-4abc=^, 

268.  By  inspecting  the  roots  of  equations  we  may 
determine  other  things,  than  when  the  roots  are  real  and 
when  imaginary;  for  example,  we  may  determine  when 
the  roots  are  equal  to  each  other. 

The  roots  of  the  equation  x'^-i-ax-\-b=0  have  already 
been  found  to  be 

Ab 


a  ,      la-  — 


Whatever  be  the  valus  of  a/ — - — we  must  ^^iits  value 
to  —  n  to  obtain  one  root  of  the  equation  and  subtract  it 
from    -^y  to  obtain  the  other  root,  and  plainly  the  only 


2^6  QUADRATIC   EQUATIONS. 

way  these  two  results  can  be  just  the  same,  is  for  the 
radical  part  to  be  0,  i.  e.  for  the  expression  under  the 
radical  sign  to  be  0.     Therefore,  in  order  that  the  roots  of 

may  be  equal  to  each  other,  we  must  have 
«2_4^=0,  or«2  =  4^. 
If  the  given  quadratic  equation  had  been  in  a  different 
form  we  could  solve  it,  and  by  inspecting  the  result,  tell 
when  the  two  roots  were  equal  to  each  other;  for  plainly, 
in  any  case  for  the  roots  to  be  equal,  the  term  preceeded 
by  the  double  sign  ±  must  be  0. 

36.  Determine  when  the  roots  of  each  of  the  equations 
in  examples  2G  to  35  are  equal  to  each  other. 

269.  Still  other  questions  about  the  roots  might  be 
answered, — as  when  are  both  rooX.s positive,  when  negative, 
when  is  one  posilive  and  the  other  negative,  etc., — but  the 
cases  given  are  enough  to  show  the  student  that  much 
may  be  learned  by  inspecting  the  result  obtained  through 
solving  a  literal  equation. 


CHAPTER  XV. 

SIMULTANEOUS   EQUATIONS  ABOVE  THE 
FIRST  DEGREE. 

EXERCISE  111. 

One  Equation  of  the  First  and  One  of  the  Second  Degree. 

270.  In  the  solution  of  simultaneous  equations  above 
the  first  degree  we  have  before  us  the  same  general  prob- 
lem as  in  the  solution  of  simultaneous  equations  of  the 
first  degree,  viz.:  to  find  by  some  combinations  of  the 
equations  given  those  values  of  the  unknown  numbers 
which  are  common  to  all  the  given  equations,  or  what 
is  the  same  thing,  those  values  of  the  unknown  numbers 
which  satisfy  all  the  given  equations  at  the  same  time. 

271.  In  this  chapter  we  confine  our  attention  to  the 
case  of  two  simultaneous  equations  with  two  unknown 
numbers. 

The  gejieral  case  of  two  simultaneous  equations  above 
the  first  degree  cannot  be  solved  without  knowing  how 
to  solve  a  single  equation  of  a  higher  degree  than  the 
second,  which  we  a-e  not  now  supposed  to  know  how  to 
do,  so  the  ge7ieral  case  cannot  be  taken  up  ;  but  there  are 
two  cases  where  the  solution  is  quite  easy.  Other  cases 
than  these  two  depend  more  or  less  upon  the  ingenuity 
of  the  student,  and  often  require  .some  special  device  for 
their  solution. 

272.  We  take  as  our  First  Case  that  in  which  one 
equation  is  of  the  first  degree  and  the  other  of  the  second 
degree. 


278  HIGHER    SIMULTANEOUS   EQUATIONS. 

I,et  US  find  the  values  of  x  and  y  from  the  equations 


x''^-y=\\ 

(1) 

X  -\-y=  5 

(2) 

From  (2),                           y=5—x. 

Substitute  this  value  in  (1)  and  we  get 

x-'-i-b-x^ll. 

Therefore,                     x'^—x=(j, 

x^-x-]-\==\\ 

^-i=±f. 

x=:3  or— 2. 
Substituting  the  first  of  these  values  in  (2), 

From  which,  y—2. 

Substituting  the  second  vahie  of  x  in  (2), 

-2-fj=5. 
From  which,  y—1. 


j  x=\ 


or 


o 


Therefore,  1         ^       - 

2  or  7. 

These  vaUies  go  together  in  pairs,  /.  ^.,  the  value  8  of  :t- 

goes  with  the  value  2  of  _>^,  and  the  value  —  2  of  x  goes 

with  the  value  7  of  ^. 

273.  If  we  now  look  to  see  just  what  was  done  in  the 
solution  just  given,  we  will  observe  that  from  the  equa- 
tion of  \.\\Qjirst  degree  the  value  of  one  unknown  number 
was  found  in  terms  of  the  other  unknown  number,  and 
this  value  was  substituted  in  the  equation  of  the  second 
degree.  This  method  is  capable  of  solving  a7iy  case  where 
one  equation  is  of  the  first  and  one  of  the  second  degree. 

Hence,  to  find  the  values  of  two  unknown  numbers 
from  two  equations,  one  of  which  is  of  the  first  and  the 
other  of  the  second  degree,  we  proceed  as  follows  : 


SIMPLE   AND    QUADRATIC    EQUATIONS.  279 

From  the  equation  of  the  first  degree  find  the  value  of 
either  unknown  number  in  terms  of  the  other  unknown 
number;  substitute  the  value  so  fou7id  in  place  of  this  un- 
hiown  7iumber  in  the  equatio?i  of  the  second  degi'ee,  and 
solve  the  resulting  quadratic  equation.  This  gives  two  values 
for  one  of  the  unknowji  numbers,  which  values  substitute  in 
tur7i  in  the  equation  of  the  first  degree,  and  thus  find  the 
values  of  the  other  unknowii  7iumber. 

Examples. 

Solve  the  following  sets  of  equations  : 


3 


6. 


(    ;«;2-|-3>/2  =  52. 

''  \  2x  -iy  =1.  9-  I 

{  ;»;2+.rj/-f^2  =  Cl.  ^^    (a'2-j/2  =  -0. 

(  x^y=l.  '   \  X  —y  =  —  1. 

•   \zx  -4y  =10.  "•    \x  -\-y  =2. 

j  x+y=G.  (  5x^—Sxy-2y'^  =  12. 

^•   I      xy=6.  "•  j  2x-y==S. 

j  x'^  +  3xy—y^=o.  (  2x'^—Sxy—4y'^  =  lG. 

'•   I  Sx-y=^l.  ^^'    \  2x-4>/=4. 

I  ;r2-j-_>/2  +  2x=31.  j    x'^— xy=S20 


2y-\-x=  7. 


M.j 


I  x-y-^l==0.  ^'   \  Sx  +12y=  12. 

xy-{-x+y=:S2.  j  xy-\-x+y=ld. 


ixy+x+y^S2.  j 


2x-\-oy=lS. 


iSx^-2xy+y^  =  81.  i  x'-+y-'-\-x+y=152. 

I  10x-2y=54:.  \  2x-y=2, 


■H 


6x'--y^-+Sx-2y=^-22D. 
4x—y=5. 


„    j  4j»;2_4>/2+.r-j/=82. 
^'''    i  4x-^4y=40, 


280  HIGHER   SIMULTANEOUS   EQUATIONS. 


ix''  +  bxy-Q^y-lx-Sy^  IG. 


(  4:X^  +  bxy- 
'9-  1  x-2yJ. 

20.    I 


10A-2-9y2+5;»r+4;/=-5G9. 
DX—dy=  —  lo. 


i  x"^  +2xy+3y^ -\-x^d(j. 
^^'    \  2x+Sy=12. 


22. 


23. 


24.   J 


25.  ^ 


f.4.- 

1 

:2. 

26.  ^ 

>/      X 

U-:r=l. 

^A--2j/=— G. 

fl--- 

27.  . 

^-fji^+-4--=G 

[;i:+ 5^=51. 

3.r-j=3. 

23.  . 

x-y^ll 

.^-J=l. 

;i;    7    ;ir    ^ 

=4. 

29. . 

^                14 

L  5^+ 5^=50. 

.  2;t:-5y^ 7. 

;ir 

1-   -^  - 

2x-\ 

:-';=-3. 

30.  . 

lx-y= 

x-y 
=  1. 

x^- 

_y-^ 

EXERCISE  112. 

Two  Quadratic  Equations. 

274.  The  Secojid  Case  is  that  in  which  each  equation 
is  of  the  second  degree  and  possesses  this  pecuHarity, 
viz  :  that  all  the  terms  which  contain  unknown  numbers 
are  of  the  second  degree  with  respect  to  those  numbers. 


TWO    QUADRATIC    EQUATIONS.  28 1 

lyCt  US  solve  the  equations 

;i:2  4-^2^25  (1) 

jr2+jry— j/2^19  (2) 

Substitute  vx  in  place  of  jk,  and  we  obtain 

^2_|_^2^2^20  (3) 

x'^-^vx'^—v'^x^  =  ld  (4) 

25 
From  (3)  we  obtain  x"^—- ;•  (5) 

19 
From  (4)  we  obtain  ^^  =  -^ (6) 

Therefore, :  = r- 

Clearing  of  fractions,  we  get 

25(H-f-z'-)  =  19(l-hu2), 
25  +  252;- 25z;2  =  i94-i9e,2^ 

44z/2_25z/=6, 

7, 2  5 1-4  1 

7, —  0  (5   f^r  16 


t/=4  or  — 


2 
TT- 


Substituting  each  of  these  values  in  turn  for  v  in  (5), 

x^± 
And  since  y==vx,  we  get 


;\:=±4  or  ±—7-^. 


jK=±3or=F-7^. 
V  o 

The=ie  values  go  together  in  pairs  as  indicated  in  the  fol- 
lowing scheme,  where  any  value  given  for  x  goes  with 
that  value  of  y  which  is  dlreclly  zinderneath  the  value 
of  X  considered  : 

^  1  11  11 

;r=4  or  —4  or      — >--  or 


Q  2  2 

_y=3  or  —3  or or      -r-^. 

"^  v'5  1/5 


282  HIGHER    SIMULTANEOUS    EQUATIONS. 

275.  If  we  look  carefully  at  the  solution  just  given 
we  will  observe  that  vx  was  substituted  for  jj/  in  each  ol 
the  given  equations,  and  from  each  of  the  resulting  equa- 
tions the  value  of  x"^  was  found  in  term-;  of  v.  The.-e  two 
values  of  x'^  were  placed  equal  to  each  other,  and  the  re- 
sulting equation  solved  to  find  the  value  of  v..  This  value 
of  V  was  then  substituted  in  the  first  of  the  equations 
which  expressed  the  value  of  x"  in  terms  of  v,  and  thence 
the  value  of  x"^  and  then  the  value  of  x  was  determined. 
The  value  found  for  x  together  with  the  value  found  for 
V  served  to  determine  j/. 

276.  The  method  used  in  solving  the  set  of  equations 
just  given  is  capable  of  solving  a?iy  two  equations  pos- 
sessing the  peculiarity  mentioned  in  the  last  article. 
Hence  the  method  may  be  stated  as  follows : 

Substitute  vx  iti  place  of  y  i7t  both  equations,  ard  from 
each  of  the  resulting  equations  find  the  value  oj  x'^  in  terms 
of  V.  Place  these  two  values  of  x"^  equal  to  each  other  and 
solve  the  resulting  quadratic  equation  to  determine  v.  Sub- 
stitute the  value  found  for  v  in  one  of  the  first  two  equations 
where  v  was  used,  and  thus  determine  the  value  of  x.  Hav- 
ing both  x  and  v,  determine  y^  which  is  the  pivduct  of 
X  and  V. 

Examples, 

Solve  the  following  sets  of  equations  : 

j  ,^2+^  =  25.  (  '■Ix'^—Zy'^^-xy^Z, 

^'   \xy=^Vl.  ^'  \Zxy-\-x'-^—'l. 

j  x'^—Zxy-Yy'^-=-h,  (  \x''-  —  hxy-Vy'^=l, 

^'  \x''-'lxy=S.  ^'    (x2-5>/2  =  _l. 

(  2x''-Zxy=  0.  (  ,r2-f  2;ri/+j/2  =  3G. 

^'  \Zxy-2y''  =  lO.  '  \  x^-2xy+y''=0. 


MISCELLANEOUS   EQUATIONS.  28^ 

I  3;»ry=60.  ^^'   \  x'^-Zxy-  y''=Z. 

i  ;t2-4>/2=0.  j  3x2-xj/=133. 


■^  ^y 


90.  '•      (  ;r>/-_y2  =  _4 


II. 


12 


^^^=^^^'  18  i-"-3-:>'-J-  =  -29. 

•  I  x""-  xy+y''  =  lS. 

C2x^-3xy-\-4y''=4.S. 
x+by= — I — 


r3 

4 

8 

Jtr2 

-7.= 

=.-^' 

j"* 

ig. 

' 

1    2 

3 

-2i 

u^ 

-y  = 

^JK 

+  3j/- 

-2a'= 

36 

> 

£0. 

. 

r>  4 

( x+y    x—y^lO 
13.  ^  ^—y    x+y~~  3  *         20.  ^ 

I  \  y'^  24 


EXERCISE  113. 

Miscellaneous  Equations. 

277.  Since  no  general  directions  can  be  laid  down  to 
cover  all  solvable  cases  of  simultaneous  quadratic  equa- 
tions, a  few  examples  will  be  worked  out  as  samples. 
The  student  is  advised  to  stud}'  carefully  the  following 
solutions,  and  try  to  see,  if  possible,  what  suggested  the 
various  steps  taken  in  the  solution. 


284  HIGHER   SIMULTANEOUS   EQUATIONS. 


(:r2+y=20. 

(1) 

^-  1  ;,  +^  =  G. 

(2) 

Square  (2), 

x^ 

+  2r;/-»-jj/«=33 

(3) 

Pit(l) 

x^ 

+;'«  =  20 

Subtract, 

2xy         =16 

(4) 

Subtract  (4)  frbm  (1), 

X* 

—2xy+y^=  4 

Extract  square  root, 

x-y=±2 

But  (2) 

x-{-y=     6 

Therefore,  adding, 

2x-H  or  4 

Hence, 

:r=4or  2 

Substitute  in  (2), 

y=^2  or  4 

(  x-hy=0. 

(1) 

2.    <                   J, 

\      xj'=b. 

(2) 

Square  (1), 

x» 

+  2r-y+y^=Zfj 

Multiply  (2)  by  4, 

Arxy         =20 

Subtract, 

x^ 

-2xy-^y^-U 

Extract  square  root, 

x-y=±4: 

But  (2> 

x-\-y=     6 

Therefore,  adding, 

2x=10or  2 

Hence, 

.r=5  or  1 

Substitute  in  (1), 

j=lor5 

ANOTHER  SOLUTION. 

hety—vx  in  both  equations; 

.  then     x  +  vx=z& 

(3) 

Z'X2=5 

(4) 

(i—x 

From  (3), 

V  — 

X 

From  (4), 

5 

G— x_    5 
Hence.  "IT^lc^ 

Multiply  each  member  by  x*.  6a;— .r^ =5 

a-«-6r=— 5 
;p«_t;x^9  =  4 

Ar-:;=±2 

iC=o  or  I 
^=lor5 

(r'+y=f?L  (1) 

^-    I  ;.T=10.  (2) 

Add  2  times  (2)  to  (1),  a:»+2  vv-h;'»=r)4 

Extract  square  root,  »-!->'=  ±8  (3) 


MISCET-LANEOUS    EQUATIONS.  28$ 


Subtract  2  times  (2)  from  (1), 

x-  —  2xy-}-y^=:4. 

Extract  square  root, 

x-y-±2 

But  '8) 

X  <-j— i8 

Therefore,  adding, 

2x=10  or  H  or  — 10  or  — C 

Hence, 

x=h  or  3  or  -5  or  —3 

Substituting  in  (2), 

^  =  3  or  5  or  -3  or  --5 

0) 

(2) 

From  (2), 

y=(S-x 

Cube. 

y 

='  =  21fi-108r+|8^2-x3 

Substitute  in  (1), 

21G 

-l(l8.r+  18^8  =  72 

Transpose  21(), 

18^*-l()8r=-lf4 

Divide  by  18, 

;r«-6r=-8 

Complete  square. 

*«-Cjr+9=l 

Extract  square  root, 

^-3=±1 

Hence, 

x=4  or  2 

and 

y-2or  4: 

ANOTHER  SOLUTION. 

Divide  (1)  by  (2), 

x«—  xy+y»  =  }2 

(3) 

Square  (2), 

x»  +  2xy+yi  =  'dQ 

(*) 

Subtract  (H)  from  (4), 

'6^y^2i 

or 

xy=S 

(5) 

Subtract  (5)  from  (3), 

x*—2xy-\-y^  =  A 

Extract  square  root, 

jc-y=±2 

But  (2) 

x+y=Q 

Adding, 

2j:  -  8  or  4 

Hence, 

x=4  or  2 

and 

;K=-2or4 

(  x^-+xy=S6. 
5-   \y-hxj'=U, 

(1) 

(2) 

Adding  (1)  and  (2), 

x^  +  2xy+y^=49 

Extract  square  root. 

x-y=  ±  7 

(3) 

Divide  (1)  by  (3>, 

^=±5 

Divide  (2)  by  (3), 

y=±2 

ANOTHER  SOLUTION. 

Factor  each  left  mem 

iber, 

x{x-i-y)=35 

(".) 

v(Xi-7)=14 

(4) 

Divide  (3)  by  (4), 

X     3 .')     5 

3^~y4~2 

Multiply  by  y. 

x=^v 

:86  HIGHER    SIMULTANEOUS    EQUATIONS. 


Substitute  in  (4), 

iy^  =  U 

Hence, 

y^=4 

Therefore, 

y=±2 

and 

x=±5 

j  x+y=(i. 

(l) 

(2) 

Raise  (2)  to  the  fourth  power, 

x^+ix^y^r6x^j 

r-i+Uy^+y^  =  U9G 

(3) 

But  (I) 

x*-i-y^=   272 

Subtract,                            4:X^y  +  Bx^jS  ^  4^^3  _  1Q24 

(4) 

Factor  (4),                           xy  2\r 

2  +  3x74  273)  =  512 

.(5) 

Square  (2)  and  multiply  result 

by  2  ry, 

Ay,2x' 

^+Axy  +  2y^}=12ry 

(6) 

Take  (5)  from  (G), 

(x/)2^72r;/- 

513 

(7) 

Transpose, 

(xy)^-l2xy=-bl2 

"Complete  square,              (-*"/)"• 

-72x)/f  129(5  =  784 

Extract  square  root, 

xy-'SG=  ±28 

Hence, 

xy  =  ij4:  or  i 

i 

(8) 

Square  (2\ 

x^  +  2xy+y^~=3fi 

Multiply  (8)  by  4, 

4xy-2J6  or 

32 

(-9) 

Subtract, 

x^—2xy  f  j-  =  — 220 

or  4 

Reject  —220  because  we  cannot  extract  the  square  root  of  —220  and 
extract  the  square  root  of  the  resulting  equations. 

x-y=±2 
But  (2)  x-\-y=     6 

Adding,  2x=8  or  4 

Hence,  x—4:  or  2 

.and  j=2or4 

Examples. 

Solve  the  following  se^s  of  equations  : 

x=^-jv^  =  63.  j  x4-y  =  G00.   . 


7 

'  X  —y  =   o 


1 


ixy+xy^^^l2. 
^'    I    x-\-xy''  =  lQ. 


PROBLEMS. 


2^ 


-I 
■'1 
■H 
-I 

■H 

■•■I 
I 


20. 


21.  ^ 


22. 


x'^  -j-j/"^  -{-x—j'=78. 
xy='n. 

^=i— j,/3  =  19(.r-j^). 

xy'^-x''y=Z2\. 
2ji:y-5A-+6>/=33. 

y'^+xy=\'{x-\-y). 

xy+x=20. 
xy—y=12. 

x^y     20* 

xy—x'^—y-  =  —dl. 
x'+y^-+xy=223. 


23 


■I 


24.   I 

-I 

26.  I 

27.  ) 

28.1 

29 


loji;>'=2520. 

x--x-hlS8=2xy+Sy 
y''-\-7y-Sx=21S. 

;r— _>/=2. 


^J/ 


-1 


30. 


31 


xy'-{-x-y-x'y'=-^^:^-g 
x-'-y^  =  21d. 
x-+xy-i-y'-=d3. 

x^-{-y^  =  lSS. 
xy(,x+y)  =  70. 

(^'4-j'-)Gr+j)  =  272. 
x^-{-y''-{-x-i-y=42. 
9 


.-2^2^ 


1. 


X- 
.rH2.r-2^)=i|^. 


EXERCISE  114. 

Problems. 

1.  What  two  numbers  are  those  whos^  product  is  24 
and  whose  sum  added  to  the  sum  of  their  squares  is  02? 

2.  Find  two  numbers  such  that  the  square  of  the 
greater  minus  the  square  of  the  less  may  be  5G,  and 
the  square  of  the  less  plus  ^  of  their  product  may  be  40. 


288  HIGHER   SIMULTANEOUS    EQUATIONS. 

3.  A  number  consists  of  two  digits  whose  sum  is  15. 
If  31  be  added  to  the  product  of  the  digits  the  digits  will 
be  in  the  reverse  order.     What  is  the  number  ? 

4.  There  is  a  number  consisting  of  two  digits,  which 
number  divided  by  the  sum  of  its  digits  gives  a  quotient 
2  greater  than  the  first  digit.  But  if  the  digits  be  in  the 
reverse  order  and  the  resulting  number  be  divided  by  a 
number  1  greater  than  the  sum  of  the  digits,  the  quotient 
so  obtained  is  greater  by  2  than  the  preceding  quotient. 
What  is  the  number  ? 

5.  Find  two  numbers  sucl^  that  their  product  added  to 
their  sum  gives  62,  and  their  sum  taken  from  the  sum  of 
their  squares  leaves  a  remainder  of  80. 

6.  A  certain  number  consists  of  two  digits  of  which  the 
digit  in  ten's  place  is  3  times  the  digit  in  unit's  place, 
and  if  12  be  subtracted  from  the  number  itself  the  re- 
mainder will  equal  the  square  of  the  digit  in  ten's  place. 
Find  the  number. 

7.  The  sum  of  two  numbers  is  37  and  the  ?um  of  their 
squares  is  greater  by  9  than  twice  their  product.  What 
are  the  numbers  ? 

8.  The  sum  of  two  numbers  is  22  and  the  sum  of  their 
cubes  is  2926.     What  are  the  numbers  ? 

9.  The  sum  of  the  squares  of  two  numbers  is  410.  If 
we  diminish  the  greater  by  4  and  increase  the  less  by  4 
the  sum  of  the  squares  of  the  two  results  is  394.  What 
are  the  two  numbers  ? 

10.  A  man  bought  some  horses  for  $1250.  If  he  had 
bought  3  more  and  paid  $25  less  for  each  horse,  they 
would  have  cost  him  $1300.  How  many  horses  did  he 
buy,  and  at  what  price  ? 


PROBLEMS.  289 

11.  A  stock  dealer  bought  some  horses  and  cows  for 
$1 1600.  The  number  of  horses  bought  was  equal  to  the 
number  of  dollars  paid  for  each  horse,  and  the  number  of 
cows  bought  was  equal  to  the  number  of  dollars  for  each 
cow.  Had  the  number  of  horses  bought  been  equal  to 
the  number  dollars  paid  for  each  cow  and  the  number  of 
cows  bought  been  equal  to  the  number  of  dollars  paid  for 
each  horse,  the  stock  would  have  cost  him  $8000.  How 
many  horses  and  cows  did  he  buy  and  what  did  he  pay 
for  each  ? 

12.  A  lady  bought  55  yards  of  cloth  in  two  pieces,  pay- 
ing $17  altogether;  for  the  first  piece  she  paid  twice  as 
many  cents  per  5^ard  as  there  were  yards  in  the  piece,  and 
for  the  second  piece  she  paid  one-half  as  many  cents  per 
yard  as  there  were  yards  in  the  piece.  What  was  the 
number  of  yards  and  the  price  per  yard  of  each  piece? 

13.  Find  two  numbers  whose  sum  is  9  times  their  dif- 
ference and  whose  product  diminished  by  the  greater  is 
equal  to  12  times  the  greater  divided  by  the  less. 

14.  Two  rectangles  contain  the  same  area,  480  square 
yards.  The  difference  of  their  lengths  is  10  yards,  and 
of  their  breadths  is  4  yards  ;  find  their  sides. 

15.  If  a  carriage  wheel  14f  feet  in  circumference  take 

1  second  more  to  revolve,  the  rate  of  the  carriage  per 
hour  will  be  2 J  miles  less.  How  fast  is  the  carriage 
traveling? 

16.  A  cistern  can  be  filled  with  water  by  two  pipes. 
By  one  of  these  pipes  alone  the  cistern  would  be  filled 

2  hours  sooner  than  by  the  other  ;  also,  the  cistern  can 
be  filled  by  both  pipes  together  in  1|-  hours.  Find  the 
time  each  pipe  alone  would  take  to  fill  the  cistern. 

19 


290  HIGHER   SIMULTANEOUS   EQUATIONS. 

17.  A  man  bought  a  number  of  shares  of  $20  stock 
when  they  were  at  a  certain  rate  per  cent,  discount  for 
$1500,  and  afterwards  when  they  were  at  the  same  rate 
per  cent,  premium  sold  all  but  GO  shares  for  $1000.  How 
many  shares  did  he  buy,  and  what  did  he  give  for  each 
of  them? 

18.  The  small  wheel  of  a  bicycle  makes  135  revolutions 
more  than  the  large  wheel  in  a  distance  of  2G0  yards ;  if 
the  circumference  of  each  were  one  foot  more,  the  small 
wheel  would  make  27  revolutions  more  than  the  large 
wheel  in  a  distance  of  70  yards.  Find  the  circumference 
of  each  wheel. 

ig.  A  sets  off  from  London  to  York,  and  B  at  the  same 
time  from  York  to  London,  and  each  travels  uniformly. 
A  reaches  York  IQ  hours  and  B  reaches  London  3G  hours 
after  they  have  met  on  the  road.  Find  in  what  time 
each  has  performed  the  journey. 

20.  A  man  arrives  at  the  railway  station  nearest  his 
home  I7}  hours  before  the  time  at  which  he  had  ordered 
his  carriage  to  meet  him.  He  sets  out  at  once  to  walk 
at  the  rate  of  4  miles  per  hour,  and  meeting  his  carriage 
when  it  had  traveled  8  miles,  reaches  home  one  hour 
earlier  than  he  had  originally  expected.  How  far  is  his 
home  from  the  station,  and  at  what  rate  was  his  carriage 
driven  ? 


CHAPTER  XVI. 

THEORY  OF  INDICES. 

EXERCISE  115. 

Meaning  of  FRACTIO^^AL  Exponents. 

278.  The  exponents  which  we  have  considered  here- 
tofore are  defined  by  the  following  equation  : 

a"=aaaa  .   .   .  to  7i  factors. 
In  other  words,  a"  is  an  abbreviated  way  of  writing  the 
product  of  n  factors  each  equal  to  a. 

279.  It  has  been  proved  that  exponents  follow  the 
five  laws  expressed  by  the  following  examples  and 
formulas : 

EXAMPLES.  FORMULAS. 

a^^  xa'^  =  a^^      Art.    39,  a" x «''=«"+''                A 

a^^-^a''=a*         Art.     54,  a"-^a''=a"-''if7i>'-rB 

(«ii)7=^77       Art.  134,  («")"=«""                    C 

{abcy=a''b''c'^  Art.  135,  {abcy=a"b''c''              D 

We  will  find  it  convenient  to  re.^er  to  these  formulas 
as  A,  D,  C  Z>,  and  E,  respectively,  and  we  will  speak 
of  them  collectively  as  The  Laws  of  Exponents  or 
Indices. 

280.  It  is  very  plain  that  the  exponent  n  must  be  a 
positive  whole  number  in  order  that  «"  may  have  a  mean- 
ing by  the  definition  in  Art.  278.  If,  therefore,  we  wish 
to  use  symbols  like  a^  ox  a  ^  we  must  first  find  a  mean- 
ing for  such  expressions. 

*  This  symbol  stands  for  the  words  "is  greater  than." 


292  THEORY   OF   INDICES. 

281.  It  is  usual  in  Algebra  to  define  the  symbols 
which  are  to  be  used,  and  afterwards  discover  what  laws 
these  symbols  follow  in  algebraic  operations.  Thus,  in 
the  first  part  of  Algebra  we  defined  positive  integral  ex- 
ponents and  subsequently  proved  that  they  follow  the  five 
laws,  A,  By  C  D,  and  E.  But  an  exception  to  this  gen- 
eral practice  occurs  in  the  case  of  fractional  and  negative 
exponents.  Here  we  first  state  some  law  which  ive  wish 
fractional  and  negative  exponents  to  follow  and  then  seek 

what  mea7iing  must  be  giveji  to  these  exponents  in  conse- 
quence of  this  law.  Thus,  we  say:  All  fractional  expo?ients 
must  follow  law  A;  required  tlie  meaning  of  fractional  ex- 
poneiits.     We  will  consider  a  few  special  cases  at  first. 

282.  Given  that  law  A  must  hold  for  fractional  ex- 
ponents; required  the  meanhig  of  a'^. 

Since  law  A  must  hold,  we  have 
a-ia'^z=a'i    i=a. 

Thus  a'^  must  be  such  a  number  that  if  it  be  multiplied 
by  itself  the  result  is  a.  By  definition  the  square  root  of 
a  is  such  a  number ;  therefore  a~^  must  be  equivalent  to 

the  square  root  of  a.     Thus  we  say 

1.       /— 
a-  =  V  a. 

283.  Given  that  law  A  must  hold  for  fractional  cx- 
ponents;  required  the  meaning  of  a'^. 

Since  law  A  must  hold,  we  have 

JL   i   i         1 4.JL  ■  1 
a'^a^a^^d^^'^^'^-=a. 

But,  by  definition  of  a  cube  root, 

V  a  \  a  V  a=a. 
Therefore,  a'^  must  be  equivalent  to  the  cube  root  of  a; 
that  is,  ^ii=  1^^«. 


MEANING   OF    FRACTIONAL   EXPONENTS.         293 

284.  Given  that  law  A   must  hold  for  fractional  ex- 
p07ients;  required  the  vieajiing  of  a^. 

Since  law  A  must  hold,  we  have 

8     3      3     3  il.».3,3 

3 

Thus  we  see  that  a'^  is  one  of  the  four  equal  factors 

which,  when  multiplied  together,  produce  a"^ .     Hence, 

3 
by  the  definition  of  a  root,  a'^  must  be  equivalent  to  the 

fourth  root  of  a^;  that  is, 

a^=V~dK 

285.  Given  that  law  A  must  hold  for  f-actional  ex- 
ponents; required  the  meaning  of  a''- . 

Let  r  be  any  positive  whole  number.     Since  law  A 
must  hold,  we  have 

1     !     -^  .  r     ,  ?  +  -4.l4-,  .  .tor terms 

a^a'^a^  .   .  .  to  r  iactors=«''   '^   '-  =a. 

Thus  we  see  that  a^  is  one  of  the  r  equal  factors  which, 
when  multiplied  together,  produce  a.  Hence,  by  the 
definition  of  a  root,  w  is  eqiiivalent  to  the  rth  root  of  a; 
that  is,  ar=:^i/a.  [Ij 

286.  Given  that  law  A   must  hold  for  fractional  ex- 

n 

po?ients;  required  the  meaning  of  a^. 

It  is  here  supposed  that  n  and  r  stand  for  any  positive 

n 
whole  numbers,  that  is,  that  -  is  any  positive  fraction. 

Since  law  A  must  hold,  we  have 

'J    ?    '-?  .  r      X.  "+"+"+•  .  .tor terms         „ 

fl'-a'-a- .   .   .  to  r  factors=«''  '-  »-  =«  . 

Thus  we  see  that  a^  is  one  of  the  r  equal  factors  which, 
when  multiplied  together,  produce  a".  Hence,  by  the 
definition  of  a  root,  a^  is  equivalent  to  the  rth  root  of  the 
71  th  power  of  a;  that  is, 

a^=^^.  [2] 


294  THEORY    OF    INDICES. 

287.  Thus,  by  requiring  that  fractional  exponents 
shall  follow  law  A,  we  have  found  that 

A77y  positive  fractional  exponent  indicates  a  root  of  a 
power,  where  the  numerator  shows  the  power  and  the  de- 
nominator the  root. 

288.  We  will  now  get  at  the  meaning  of  a  fractional 
exponent  by  a  little  different  process,  but  still  by  requir- 
ing that  such  exponents  shall  follow  law  A. 

From   the   meaning   of  a  positive   integral   exponent 

(Art.  278)  we  know 

.1-.         Ill 

(a^y'=a>'a>-a'-  .   .   .  to  n  factors. 

T,      1  yf  l-fl  +  l-f ...  to « terms 

By  law ^,  =«r^r^»--r 

Adding  the  fractions,    =«'-. 
Therefore  we  have  shown  that 

i  ?! 

(«>)«=«'•,  [3] 

or,  since  a^^l^a,  by  [1],  we  have  shown  that 

«"=(]/«)«;  [3] 

That  is,  a^  is  cqinvalent  to  the  n  ih  power  of  the  rth  root  of  a. 

289.  Thus  we  have  shown  that 

Any  positive  fi^actional  exp07ient  indicates  a  power  of  a 
root,  where  the  nu^nerator  shows  the  power  and  the  denomi- 
nator the  root.  ■ 

290.  Comparing  this  wdth  Art.  287,  we  see  we  have 

n 

found  two  inea7iings  for  a^ \  first,  the  rth  root  of  the  n  th 
power  oi  a  ;  second,  the  w  th  power  of  the  rth  root  oi  a. 
From  equations  [2]  and  [3]  we  get  this  statement  in  the 
form  of  an  equation  as  follows  : 

a7-=v^/^=(^^)'V  [4] 

or,  writing  exactly  the  same  equation,  but  using  frac- 
tional exponents  instead  of  radical  signs, 

a"=(a'0^  =  («b^  [4] 


EXAMPLES.  295 

n 

While  we  have  found  tzvo  772eanings  for  d^  yet  these  two 
meanings  always  give  the  same  mtmerical  results,  as  is 
proved  by  equation  [4].  Thus:  8^=l>'82  =  (f/8  ;2, 
81T=^''812  =  (f/8i)2,  etc.  The  order  of  the  operations 
is  different,  but  the  results  are  the  same. 

291.  The  above  truth  is  very  important  and  may  be 
stated  in  the  following  manner : 

The  rth  root  of  the  71  th  poiver  of  a  number  is  equal  to  the 
n  th  power  of  the  r  th  root  of  that  number. 

EXERCISE  11c. 
Examples. 

Write  each  of  the  following  sixteen  expressions,  using 
fractional  exponents  in  place  of  radical  signs  : 

I.   l/rt.  5.   V~a}.  9.   l^^^        13.   f/«— 5. 

2.  ]/?.      6.  o/ay.  10.  {fxy.  14.  {C^^x^yy. 

3.  i^7\      7.  f/^.      II.  1^?.       15.  ^'^^^zHi 

4.  #^^.      8.  if  ay.  12.  {\^xy.    16.  i'\a+xy. 

Find  the  numerical  value  of  each  of  the  following  six- 
teen expressions  : 

17.  4i  21.  6254-.        25.  8li  29.  25Gi 

18.  273-.        22.  Gli  26.  I25I        30.  64^ 

19.  oi  23.  21Gi        27.  32!  31.  5123-. 

20.  IGT.        24.  IGt  28.  814-.  32.  1287-. 

•  Write  each  of  the  following  expressions  /;/  tzuo  ways^ 

using  radical  signs  i;istead  of  fractional  exponents: 
1  2  7  « 

33-  dS.  37.  71^.  41.  ?-8.  45.  «». 

34.  15".  38.    b^.  42.   Jtv.  46.   ^^i". 

S  5  1  !£±_1 

35.  ;;z^+".  39.  e^K  43.  J'^  47-  -^  '•'• 
35.   x^.             <o.   /^"^.           44.   t:^-^.  48.   a' t  . 


296  THEORY   OF   INDICES. 

EXERCISE  117. 

Properties  of  Fractional  Exponents. 

292.  To  prove  that  a  number  with  afractiotial  exp07ient 
has  the  same  value  whether  the  exponent  is  expressed  in  its 
lozvcst  terms  or  not. 

Let  71,  r,  and  /  be  any  positive  whole  numbers.  We  are 
to  prove  that  a^^^a'-t. 

We  know  that  a=  {cc'^y' 

because  the  rt\\\  power  of  the  r/th  root  equals  the  num- 
ber itself.     Taking  the  rth  root  of  each  side, 

Raising  both  sides  to  the  n  th  power, 

Each  side  is  now  a  power  of  a  root.     By  the  meaning  of 
fractional  indices  (Art.  289)  we  write  this 

n  nt 

a>-~a'-t^  [5] 

which  is  what  was  to  be  proved. 

293.  Having  found  a  meaning  for  fractional  exponents 
by  requiring  that  they  follow  law  A,  we  will  now  prove 
that  they  also  follow  laws  B,  C,  D,  and  E. 

294.  To  prove  that  fractional  expone^its  follow  lata  B. 
Let  n,  r,  s,  and  /  be  any  positive  whole  numbers,  so 

7t  S 

that  —  and  -  are  any  positive  fractions.     We  will  prove 
^'•-T- «'=«'•   ',  if  ^>7. 

n  s  tit  sr 

We  know  a'--7-<2^=«^/-=-«'^^,  by  Art.  292. 

=  («^0'"-^(«^)'">  by  equation  [4j. 

=  (^^0"'~",  by  law  B  for  integral 

indices,  since  nt  and  sr  are  whole  numbers  and  nt^  sr 

if  '^>\. 

r     t 


PROPERTIES   OF    FRACTIONAL   EXPONENTS.      297 


ftt—sr 


=a-^,  by  Art.  289. 

But  this  last  fractional  expouent  is  what  we  would  get  if 

we  subtract  -  from  — 
/  r 

Therefore,  a'--T-a'=a'-~f,  if  ^'>7,  [6] 

which  is  what  was  to  be  proved. 

295.   To  prove  that  fractional  expone^its  follow  law  C. 
Let  «,  r,  s,  and  /  be  any  positive  whole  numbers. 

n  us 

Case  I.  We  will  prove  («^) '=«'•". 

n  n    n    n 

We  know         {wy^a^a'^a^  ...  to  ^  factors, 

by  definition  of  an  integral  index. 

=.a»>---'°''^''''\  bylaws. 

=  «'-,  by  adding  fractional  indices. 
Therefore,  {aly=a'^,  [7] 

which  is  what  was  to  be  proved. 

Case  II.  We  will  prove  (a^)''=d:>', 

n    1  tit     1 

We  know  («')'  =  («^0'",  by  Art.  292. 

=  ([«-]"')'•. 
by  meaning  of  a  fractional  index. 

=(1[''"3"!0'. 

by  law  C  for  integral  indices. 

Dy  taking  the  t  th  root  of  the  /  th  power. 
ft 

by  meaning  of  fractional  index. 
Therefore,  {a*!y=a^t^  [8] 

which  is  what  was  to  be  proved. 


298  THEORY   OF   INDICES. 


s  ftr 


Case  III.  We  will  prove  (a^y^ast, 

Wekuow  («'0^"=[(«")0'. 

by  llie  111  calling  of  a  fractional  index. 

=  l^"'Y,  by  Case  II. 

tis 

—  a'-',  by  Case  I. 

Therefore,  (a-y^^^ar^  ,  [9] 

which  is  what  was  to  be  proved. 

296.  To  prove  that  fractional  exponents  follow  law  D. 

n  ft    n  M 

We  are  to  prove  (abcy^a^b^c^. 

We  know     {abcy^-liabcyy , 

by  meaning  of  a  fractional  index. 

by  law  D  for  integral  indices. 

by  law  C,  (Art.  295,  Case  I). 

by  law  D  for  integral  indices. 

«    n   tt 

=a~^b'-c^y 
by  taking  the  rth  root  of  the  rth  power. 

Therefore,        {abcy=a~b'^c'^,  [10] 

which  is  what  was  to  be  proved. 

297.  To  prove  that  fractiorial  exponents  follow  law  E, 
We  are  to  prove  (j\ ''=—^-, 

WeUnow  g)"=[G)7 

by  the  meaning  of  a  fractional  index. 


EXAMPLES.  299 


{?)' 


by  law  B  for  integral  indices. 
«       1 


by  law  C  (Art.  295,  Case  I). 


=[(3)T 


by  law  B  for  integral  indicCvS. 

n 

by  taking  rth  root  of  rth  power. 

n 

Therefore,  (3'=-^  ["] 

which  is  what  was  to  be  proved. 

EXERCISE  118. 

Examples. 

Perform  the  indicated  operat'ons  in  each  of  the  fol- 
lowing examples  by  means  of  the  laws  of  fractional 
exponents  just  proved  : 


I.  a^Xd^. 


2        « 
2.  a^Xa^'. 


Jxa^=a^'^hhy  A)=J. 


3.  a^Xd^.  7.  d^Xd.  II.  a-l?^x2a^d'S. 

1      .         ,36 


4. 

x'^  X  x^. 

8. 

111^X111^. 

12. 

3jf"^>*  X  2.r^^T 

5. 

xixxi. 

9- 

.ix^i 

13. 

a^"Xa^". 

6. 

1         s 
c^Xc^. 

10. 

<^^x/^i 

14. 

2               r 

a"  X  d^". 

300  THEORY   OF    INDICES. 


15. 

at^at. 

aK 

a^=z  J~\hy  B)=a^=> 

a^(by  Art.  292). 

16. 

dT^di. 

i*. 

r-d^=d^~^{hy  B)z=i^^- 

-^^=^^. 

17. 

a^-r-ai. 

21.    xi^x'^. 

3w                M 

25.    «  '-  -^flS'-. 

i5. 

/^t-^/^i 

22.  6A't--3j»;2-. 

26.  8a5/^^~4«2^i 

19- 

kUkk 

23.    771^-— 771^. 

27.  6x^ji^Sxijy^. 

20. 

24.  9a^-7-a^. 

28.  ahTr—a^b^r, 

29. 

(a^)i=rz~iV(by  C), 

30. 

(di)i. 

i^h 

^-<^^'"(by  C)  =  ^^'<^(by 

Art.  292). 

31. 

(^*)i. 

35.    (J'¥. 

39.    (-^^")^^ 

32. 

(«*)A 

36.  («*)i 

40.    [(^-)"]^. 

33- 

(.4)i 

37.    (^¥. 

41.  [(^'01". 

34- 

(/i^)*. 

38.    («tV)4. 

St      3 

42.    (;ir6''«)^ 

43.  {(^bcy. 


44.   (a4.r2jK«)^. 


(ff^O*=«^<^^Aby  -Oj. 


'2     1     .S     fi 

45.  (a3^2^T)¥. 

(aMc-^)»=it(a^>^(/5^)^(^^;°(by  Z>)=aM/^(by  C). 

46.  (aH^)i.  50.  (jcl 2^6)1  54.  («l^f^^)i 

47.  («^/^^)V".  51.  (2^.r^_>/t)i2.  55.  (8«6^M)I 

48.  (aV^)i  52.  (32jt:tj/f)t  56.  {S6a^x"jy^y 

49.  {a^b^)'^.  53.  (a4^.;»;^_>/i)i*.  57.  (i^^^'^r^-^s. 


EXAMPLES 

• 

^  (if- 

^^(by  C). 

»-  G)*^ 
-  &■ 

-  (5f 

^  (7)' 
-(f 

-  ii)'- 

"■  iff- 

67.     (-2V 

68.  (a^+fl*4-l)(«^4-^-«^). 

We  arrange  the  work  thus  : 

a^^a^  +  a^ 

n^ 

..11 

J 

30] 


69.  (;r+jt'7j'7+j')(^— -^Ir^+J'). 

70.  Ci:+2y^-f-3)/i)(;tr-2y^"  +  3>/^). 

71.  («t4.«i^i+^l)(ai— /^i). 

^  R  S  R 

72.  (x-^-hj^'^){x^—jy'^). 

73.  (-r^-^+^'^-l)(-^+.'r^+l)- 

75.   (_x^-xY^+/Xx^+xy^-hy^). 


302  THEORY   OF   INDICES. 

76.  {(^—2aP^Zc^'){^aP—aP). 

77.  i^abi—Za^b^"-.    ^ 

78.  (xi—xj/i-\-x^jy—yi')-^(xi—yi). 
We  arrange  the  work  as  follows  : 

i         I 

It  is  just  as  important  to  keep  dividend  and  divisor  arranged 
according  to  the  powers  of  some  letter  in  case  the  exponents  are 
fractional  as  in  the  case  they  are  integral.  The  fractional  and 
integral  exponents  must  take  the  order  of  their  respective  magnitudes. 

79.  {x^—x'^—Axi+Qx-2x^)-^{xi—Ax^-\-2'). 

80.  {x-\)-^{x^-\). 
Si.  (J,_l)-^(_yi_l). 

82.  {a—b''-^-^{a^-\-aib^-^a^b  +  bi). 

83.  («i— 2^?tjt-t-f-ji:3)^(^T_2«lxi+x). 

84.  {x+y-]-z~  Zx^j^z^^)  ^  {x^  -^y^ + z^). 

85.  {a^-b^—c^-\-2b^c^')-^{aJ+b^-c^i). 

•86.   (Sx^ + ^i —z  +  Q>x^y^z^)-^{2x'^ + y^ —z^^. 

87.  Find  square  root  of  x'^  +  2x'^-\-\. 

1  11        1 

88.  Find  square  root  of  4a'^—4.r 3  j/T-j-j/T. 

89.  Factor  jr— 2.1-^1^^ +j'. 

90.  Find  square  root  of  Jt't—4.v^+4;r+2.r6—4.r^+.r'3". 


ZERO   AND    NEGATIVE    EXPONENTS.  303 

EXERCISE  119. 

Meaning  of  Zero  and  Negative  Exponents. 

298.  If  the  product  of  two  numbers  is  unity,  either  of 
the  numbers  is  called  the  Reciprocal  of  the  other  number. 
Thu^,  i  is  the  reciprocal  of  2,  |-  is  the  reciprocal  off,  etc. 
In  other  words,  the  reciprocal  of  a  number  is  1  divided 
by  that  number. 

299.  Given  that  negative  exponents  must  follow  law  A; 
required  the  7neaning  of  a'"^ . 

Since  law^  must  hold,  we  have 
Therefore,  by  dividing  both  sides  by  a', 

That  is,  a- 2  =  —^. 

a" 

Hence,  a~'^  is  equivalent  to  the  reciprocal  of  a^ . 

300.  Given  that  negative  exponents  must  follow  law  A; 
required  the  meaning  of  a~'^. 

Since  law  A  must  hold,  we  have 

aa~^=a^~^=-d^, 

_^     rt^      1 
Therefore,  c    a==  — =_-; 

^      at 

-%      1 
That  is,  a   3=-^. 

'a^ 
_i  2 

Hence,  a  ^  is  equivalent  to  the  I'ecipi'ocal  of  d^ . 


304  THEORY    OF    INDICES. 

301.  Given  that  negative  exponents  must  follow  law  A; 

requii'ed  the  meani7ig  of  a~". 

Let  n  be  any  positive  number,  integral  or  fractional,  so 
that  —71  stands  for  a  negative  whole  number  or  fraction. 
Since  law  A  must  hold,  we  have 

Let  r  be  taken  greater  than  n.     Then  we  know 

a"  ' 

a" 
Therefore,  a^'a'" = --• 

Dividing  both  sides  by  a",  we  obtain 

Hence,  a~'*is  equivalent  to  the  reciprocal  of  a**, 

302.  Since  n  stands  for  any  whole  number  or  fraction 
in  the  above  work,  we  may  say  : 

Any  number  with  a  negative  exponent  is  equivalent  to 
the  reciprocal  of  that  number  with  the  same  exponent  taken 
positive. 

303.  Given  that  zero  exponents  imtst  follow  law  A;  re- 
quired the  7neaning  of  a^ . 

Since  law  A  must  hold,  we  have 


r«/T  0  =  /,"+  0 


=a' 


Therefore,  a^  =  l. 

That  is,  a^  is  equivale7it  to  unity. 

304.  Since  a  in  the  above  work  stands  for  any  nuniDer 
whatever,  positive  or  negative,  integral  or  fractional,  we 
may  say  : 

A7iy  7iumbcr  with  the  expo7ient  zero  is  eqiiivalent  to  unity. 

Thus,  20  =  1,  30  =  1,  5«  =  1  100  =  1,  A'0  =  1,  (xr)°  =  l, 
(2«  +  4.y2)o  =  1,  etc. 


EXAMPLES.  305 

305.  It  follows  necessarily  from  Art.  301  that 

,  s.-t_a"b-''c' _a"c'_     a" \ _b-''d-' 

"^  ^    ""  d'         fd'     c-b'd'     a-"b''c-^d'~a-"c-' 

That  is  :  Afiy  factor  7nay  be  transferred  frorn  07ie  ierni 
of  a  fraction  to  the  other  term  provided  the  sign  of  the  ex- 
ponent of  that  factor  be  changed. 

2-Kv     b\r    4r^a-H''     S^bH""      ^ 

Thus,  rr-7i:^  =  9T-'    — i ="1 »  ^^C-'  ^^C- 

oab         2U     8-4^2^-2     4i^^2 


EXERCISE  120. 

Examples. 
Find  the  numerical  value  of  earh  of  the  following  : 


I. 

20. 

5. 

10-^ 

9- 

2(a  +  by 

13. 

1024"^ 

2. 

2-1. 

6. 

1-^ 

10. 

16-1 

14. 

512-i 

3. 

3-*. 

7. 

1-". 

II. 

G4-i 

15- 

G4-t 

4. 

(-2)-3. 

1 

2«>' 

8. 
20. 

8a^2-\ 

12. 

81-i 

16. 
26. 

G25-i 

17. 

5 

23. 

5-2 

IG-J 

(-4)-=^" 

3G-^' 

50  • 

18. 

1 
1-1' 

21. 

2-4 

4-2- 

24. 

27-i 
2-3-- 

27. 

9-2 

81-f 

19. 

2 

0-'>* 

0 

22. 

1-8 

8-1- 

25. 

32-^ 
2"!  ' 

28. 

70 
49- i' 

Write  each  of  the  following  expressions  without  using 
zero  or  negative  exponents  : 

29.  x^.  32.  oa-K        35.   (x+j-y.  38.  2«^jir-2^~i 

30.  «-^         33-  G^2^-\     36.  A-O-fj'^    39.  a^b~ie-'d'K 


^         ,._! 


31.  x2_y--.    34.  Sa-H-^\S7.   i-x)-K  40.   (-«-)-^ 

23 


3o6 


41.  ^4--        45.   ;:33.  49.    [-)  •       53- 

42.  -TT.  4e>.   ^TT-r^.    50.  -0.  54. 


X' 


in 


THEORY   OF 

INDICES. 

45.    ^-3-          49. 

©•■ 

4^-   .A2.-3-    50. 

1/0' 

1 

«~2 


3«tri 


b-'' 

c'' 

a 

-bi 

Zx- 

-Vi' 

'  Za' 

0^-2^-4 

5a- 

H-^c-^' 

9a 

^x~iy-^ 

43.  -;^.       47.  ;^pi-r^.    5i.  -— j--    55. 

^  ^    y  ba    ^b 

44.  — ,-.  48.    — — r.        52.    z,— 56. 

Write  cacli  of  the  following  expressions  in  one  line : 

„     4  ^      3.r^j/-3  -       x''y\a-by 

c  a  u  2  r3^-2_^,  -5 


(«— Q    ^<;    '2 


jy     X     X' 


EXERCISE  121. 

Properties  of  Negative  Exponents. 

306.  We  have  required  that  negative  exponents  follow 
law  A  in  order  to  obtain  a  meaning  for  them.  It  i.^  pos- 
sible to  prove  that  negative  exponents,  having  the 
meaning  thus  found,  also  follow  laws  B,  C,  D,  and  B  in 
the  operations  of  Algebra. 


PROPERTIES    OF    NEGATIVE    EXPONENTS.         307 

307.   To  prove  that  negative  expone7iis  follow  law  D. 
lyCt  n  and  r  stand  for  two  positive  numbers,  integral  or 
fractional;  then  —71  and  —rare  any  two  negative  numbers. 

Case  I.  We  will  prove  «" -7- «-''=«" "(-''). 

We  know  a"-^a~''=^"  X  -^- > 

a 

by  properties  of  fractions. 
=^"x  «-(""''), 

by  meaning  of  negative  index. 
=«""("''),  by  law  A. 

Therefore,  a" -^rt-''= «"-("').  [13] 

which  is  what  was  to  be  proved. 

Of  course  —(  —  /')  can  be  written  ■\-r,  and  n—{—r)  can  be  written 
M  +  r.  The  form  n — (  — /')  is  kept  merely  to  show  the  subtraction  of 
the  negative  index. 

Case  II.  Wc  will  prove  a~"—a''=a~"~'', 

Wc  knov/  a~"-T-a''—a~"x—^ 

a 

by  properties  of  fractions. 

=a~"Xa~'', 

by  meaning  of  negative  index. 

=a~"~'',  by  law  ^. 

Therefore,  a-"-^a''=a-"-''.  [14] 

which  is  what  v/as  to  be  proved. 

Case  III.  We  will  prove  «-"-T-a-''= «-"-(-''). 

We  knov/        a~"~a~''—a~"  x  -—> 

a 

by  properties  of  fractions. 

—a-"Xa-^-''\ 

by  meaning  of  negative  index. 

—  a~"~^~''\  by  law  A. 

Therefore,  a-"-^a-''=a-"-^-''\  [15] 

which  is  v;hat  v.'as  to  be  proved. 


308 


THEORY   OF    INDICES. 


308.  It  should  be  noticed  in  the  above  demonstration  of 
law  D  that  na  restriction  whatever  is  placed  tipnn  the  relative 
7nagnitudes  of  the  exponents  n  and  r.     Consequently, 

Law  B  is  proved  for  all  kinds  of  exponents^  whether  n  is 
numerically  greater  than  r  or  7iot. 


Thus:     a^-^a^=^a 


5_/,-2 


■^a^^a' 


etc. 


309.   To  p7vve  that  negative  exponents  follow  law  C. 
Let  n  and  r  stand  for  any  two  positive  numbers,  in- 
tegral or  fractional. 

Case  I.  We  will  prove  {a")~''=a~"''. 

We  know  («")"''=  t—^jti' 

{a  ) 

by  meaning  of  negative  index. 

by  law  C  for  positive  indices. 

by  meaning  of  negative  index. 
Therefore,  («")"''=  ^""^  [16] 

which  is  what  was  to  be  proved. 

Cask  II.  We  will  prove  (<2-") "=«"'"'. 

We  know  («-")'  =  (^-)  , 

by  meaning  of  negative  index. 
1 


Therefore, 


by  law  B  for  positive  indices. 


=a' 


by  meaning  of  negative  index. 
(a-r^a-'"-.  [17] 


which  is  what  was  to  be  proved. 


PROPERTIES    OF    NEGATIVE    EXPONENTS.         309 
Cask  III.  We  will  prove  {a-")-''=a"\ 
We  know  {a-y^-^^. 

by  meaning  of  negative  index. 


=-— -,  by  Case  II. 

a 


=  «"% 
by  meaning  of  negative  index. 

Therefore,  (0-")-"=^""'  [18] 

which  is  what  was  to  be  proved. 

310.  To  prove  that  negative  exponents  follcw  law  D. 
Let  n  be  any  positive  number,  integral  or  fractional. 
We  will  prove  {abc)~*'=a~"b~"c~". 

We  know  (fl^r)-"= 7-i- . 

^      ^         {a  bey 

by  meaning  of  negative  index. 

^     1 

'^a"b"c"' 

by  law  D  for  positive  indices. 

a"  ¥  d' 

by  properties  of  fractions. 
=a~"b~"c''*'y 
by  meaning  of  negative  index. 
Therefore,  {abe)-"^a-"b-"e-\  [19] 

which  is  what  was  to  be  proved. 

311.  To  prove  that  7iegative  exponents  follow  law  E. 
Let  n  be  any  positive  number,  integral  or  fractional. 
We  will  prove  y~\     =y:,r 


3IO 
We  know 


THEORY    OF    INDICES. 


(y  )     = -^^  by  meaning  of  negativ 


index. 


— ;;»  by  law  E  for  positive 

fL  indices. 

0'' 


b'' 


by  reducing  fraction. 


b-" 


by  meaning  of  negative  indices. 

Therefore,  G)"=i^'-  PO] 

which  is  what  was  to  be  proved. 


EXERCISE  122. 

Examples. 


Perform  the  indicated  operations  in  each  of  the  follow- 
ing examples  by  means  of  the  laws  of  exponents,  now 


proved  to  hold  for  negative  and  fractional  exponents. 


1.  «^  xa  ^. 

2.  b-^xb-^. 

3.  c''  Xc-^. 

4.  d^  Xd~^. 

5.  ;ri2xr-i 


6.  ?^~^^  X7V 

7.  a~^xxa~''j. 


_  1-         4 


8.  ar^xxa^x'^.   13.  ^    ^X^~^. 


9.  8a 


10.   G^Ji:"^'  x^^A'-.    15. 


_  2.        _i 
14.   7;e    a  X  w    ^. 

3 


?^ 


-1 


Xzci. 


16.  Ga   '■xx2^-'-x~^. 

17.  C-7a-H--)<i--4aH-^)(i-aH''x-''), 

18.  (2^^/^~S)(^-|-/,l—i_j7,f+^f /,-!). 


19.  «^-r-<2""**. 

20.  b-*^b-^. 

21.  ^:-S-r•^~^. 


22.  «-^-^-«ll. 

23.  ^-5-^^-11 

24.  :r^-T-Ji:~ii. 


25.  rt-2^=^-T-a4^-^ 

26.  ab-'^-^ar^b^. 

27.  a~^b~"-^ar^b- 


EXAMPLES.  3  I  I 

23.  2«o^-V-2-^3«-ir2j»rO. 
2g.   Za-^-y-c^-^ha^'b-^c-"-. 

30.  7^-^^-2^-3-^-8^-2/5-3^-* 

31.  5Gx5j/-7^*^7;«r-^jK-^^-*. 

32.  18«~2Z>^r-5^Ga^<5^<:-5. 

33.  6;i:^j/   ■S[y6-i-2jr  ^^i/^^-   s". 

34.  (ai(^+3fl<52_^5^f^:j)_j.(^i^)^ 

35.  («-')-^        44.  (-^Vi  53.  («--^'')i 

36.  {a-^y.        45.  (r-«)J  54.  (.r-54y^. 
37..  («-')^        46.  (^-^ri       55.  («2^-^-i 

38.  (a-2)-\  47.     («/5^)-4.  5G.     (^-5/5-10)-! 

39.  («-')-'.  48.   {a-bc--)-"-'.  57.   (-Lr"j-9)i 

40.  (a^)-2.  49.   {a-^b''c^Y'\  58.   (-Ix'')-^ 

41.  {n-^y,  50.   (7«-6x-o)-2.  59.   (-a-3)4. 

42.  («i)-3.  51.   (jrVV'^.  60.   (_«-i)-3^ 

43.  (^~V*.  52.  (8i3^-«)-i  61.  (-««)-! 

G2.   (— Srt-^/^J^:"^)"'^. 


=«■  (5)"'     <*■  (?)--     -  C-^)* 

''■("f;)"   -(T^r*  -er 

^^   (7^)  71.  (^.r^--,)        76.  (3;^^) 


3X2  THEORY   OF    INDICES. 

79.   (iax-^-{-dx-^+cx-^+dx-^-j-cx^)xx'^. 
Co.   (a-jr-i+3«";»;-2)(4a-i-5.r-i+Ga;t:-2). 


Ci.  (2x"'^—  3.v0  +4A-o)(A-~t— 2.r~i+3.r~i), 

-3  -8  -I 

X   "—2-1-    "  ^-  3r    » 


— i  —^  _J 

2jr    "— 4r    »+  Gr    " 

-3  -3  _l 

—  3x    "-I-  Gr   "—  Ox   » 

4.r~g—  8.r~"  +  12rO 

G2.   (.r-^2-+_y-2)(^^-i__^-2>)^ 

83.  (;r4jj/+j/t)(;e-i__y--r)^ 

84.  {x-\-x^+x~^s'){x^6^x-^—x-'^). 

85.  (•^~*+^"^+l)(;r-i-l). 

86.  {x-^+x-i+l)(^x-''-x-i-l), 

87.  (a'~J-2jr~t;/t-fj,i)(;j;-i_^l). 

88.  (-r-t_x-i+,i;-i_i)(^-i^l)^ 

89.  (2ai-Zax^)Q}>a-  '^+2x-^){4a^x^-^^a-^xi). 
gi.   (^x-i- x-'^y^-^x-^^y-yiy^^jc-^'-yt), 

x-^-yl  )  x~^-x--^y^-\-x-'^y-y^[    x-^ +y 


■^-X-^yh 


xjy-y^ 
x~^y—y^ 


EXAMPLES.  313 

93.  (;»;-i— j/-i)-f-Or"^— j)/~3). 

94.  (;»;-3+2.r-2-3jtr-i)^Cr-2  +  3;»:-i). 

Arrange  terms  according  to  powers  of  x,  so  that  the  exponents  of 
dividend  will  be  in  the  descending  order,  4,  2,  0,  —2,  —4. 

96.  {x~'^—x-''-  —  ^x'^^-^x-^ _2^-^)-i-(;r"l— 4ji:"^4-2) 

97.  {x^-\-1xi-\-\—x~^)-^{x^^-x~^—x-'^). 

98.  {x^—x    ■^^-^{x'i—x   2). 

99.  Simplify  ^^ <—^.  100.  Simplify  -^^ — \^       '  . 

(^"')~^  («;r)-2i/7ira 

loi.  Simplify  [(rt"J^i)--^x  («~^^~^p]-2  4. 
102.  Simplify  [^^2(;^^3)i(^2^3yl^-]5-. 
103.  Simplify  (2i"a4-3^(3ifl-2^-GT(«2__^2)^2ia^. 


CHAPTER  XVII. 

SURDS. 

EXERCISE  123. 

Definitions  and  General  Principles. 

312.  From  the  last  chapter  the  student  has  learned 
that  there  are  two  methods  in  use  for  indicating  the  root 
of  an  expression,  one  by  the  ordinary  radical  sign  and 
the  other  by  a  fractional  exponent.  While  it  is  unnec- 
essary to  have  two  ways  of  writing  the  same  thing, 
yet  eaob  method  of  notation  has  special  aciv^antages  in 
particular  cases,  which  accounts  for  the  use  of  the  two 
methods.  Of  course  the  same  laws  (namely,  A,  B,  C, 
D,  and  E  of  the  last  chapter)  govern  the  operations  with 
roots,  whatever  form  of  notation  be  used. 

313.  When  a  root  of  an  arithmetical  numeral  can  onlj?- 
be  found  approximately,  that  root  is  called  a  Surd. 
Thus,  |/2  and  T^  5  are  surds.  Expressions  like  l/4, 
1/8,  etc.,  are  said  to  be  in  the  form  of  a  S7ird.  Expres- 
sions like  l/«,  'V ab,  etc.,  are  often  called  surds,  although, 
of  course,  they  are  only  such  when  the  letters  stand  for 
numbers  whose  roots  cannot  be  exactly  taken. 

314.  Surds  are  of  the  same  Order  when  the  same  root 
is  required  to  be  taken  in  each.  Thus,  l/2  and  Vo  are 
of  the  Second  Order,  1^  3  and  f^2  are  of  the  Third 
Order,  etc.  Surds  of  the  second  and  third  orders  are 
often  called  Quadratic  and  Cubic  Surds  respectively. 


DEFINITIONS   AND    GENERAL    PRINCIPLES.      315 

315.  Monomials,  binomials,  trinomials,  etc.,  which 
contain  surds  are  called  respectively  Monomial  Surds, 
Binomial  Surds,  Trinomial  Surds,  etc.  Thus,  Sf  5 
is  a  monomial  surd,  2-f-l/5  is  a  binomial  surd,  and 
1/3  —  1/2  +  1/5  is  a  trinomial  surd. 

316.  A  factor  written  before  a  surd  is  often  called  the 
Coefficient  of  the  Surd.  Thus,  in  2a]/ d,  2a  is  called 
the  coefficient  of  the  surd  V  b. 

317.  Any  expression  containing  surds  is  called  an 
Irrational  Expression,  and  any  expression  not  con- 
taining surds  is  called  a  Rational  Expression. 

318.  The  operations  with  surds  depend  upon  prin- 
ciples established  in  the  last  chapter.  For  convenience 
of  reference,  we  will  restate  below  those  principles  which 
we  shall  make  use  of  in  the  present  chapter. 

319.  The  rth  root  of  the  product  of  several  numbei's  is 
equal  to  the  product  of  the  rth  roots  of  the  several  numbei's. 

That  is,  V  atjc=^VaV'bV~c,  [1] 

1       1  1.  1 
because  {abcy—a'b''c\  * 

by  equation  [10],  Chapter  XVI. 

320.  The  rth  root  of  the  qitotient  of  two  mivibers  is  eqiiat 
to  the  quotient  of  their  r  th  roots. 


V  b 
because 


by  equation  [11],- Chapter  XVI. 


3IO  SURDS. 

321.  The  rtth  root  of  a  nitmber  equals  the  rih  root  of 
the  t  th  root  of  the  ruimber. 

That  is,  Va=^  ^  i/a,  [3] 

because  a'''=^{a^y^ 

by  equation  [9],  Chapter  XVI. 

322.  The  rtth  root  of  the  ntth  power  of  a  ntmibcr 
equals  the  rth  root  of  the  nth  pozver  of  that  munber. 

That  is,  V'^'^V^'  [4] 

because  .  a>i~a'-, 

by  equation  [5],  Chapter  XVI. 

323.  The  n  th  power  of  tJie  r  th  root  of  a  number  equals 
ihe  r  th  root  of  the  n  th  power  of  that  number. 

That  is,  {i/'ar=\/~a\  [5] 

This  is  equation  [4],  Chapter  XVI. 

EXERCISE  124. 

To  Remove  a  Factor  from  Beneath  the  Radical  Sign. 

324.  When  any  factor  of  the  number  under  the  radical 
sign  is  an  exact  power  of  the  indicated  root,  the  root  of 
that  factor  may  be  extracted  and  written  as  the  coefficient 
of  the  surd,  while  the  other  factors  are  left  under  the 
radical  sign. 

(1)  Thus,  l/8=l/'4x2_ 

=  l/4l/2  by[l]. 

=  2l/2 
<2)  Also,  f^81  =  f/27x3 

=  r27)/3  by[l]. 

==3l>'3 

(3)  Also,  \/\{jax^  =  jySx^  x  2ax 

■    =\y8x-'f/2ax  by[l]. 

=  2x^2ax 


TO   INTRODUCE    COEFFICIENT.  317 

Examples. 

Remove  as  many  factors  as  possible  from  beneath  the 
radical  sign  in  each  of  the  following  : 

1.  1/12.         9.   1^5G.         17.  5tlU.    25.   f/a^-'d'^ 

2.  l/28.       10.   ^TgO.       18.   1/^.      26.   1/4^. 


3.  1/50.       II.    ^2048.     19.   f/;;rU'^  27.   V2om*x. 

4.  1/72.       12.   1^8645.    20.   l^-^^s?.     28.   l/3G«-'^^ 

5.  #^72.       13.   f'5G7.       21.   1^^.       2Q.   f'a^xy. 


6.  1^^500.     14.   1>''112.      22.   f'c'^x\  30.   1^81;;/«x-. 

7.  1^108.     15.  21/^405.    23.   f'x'y\    31.   l^G4a8/;«. 

8.  TKiu2.    16.  3P^8G4.    24.  l/y*7^    32.  2l/8(W. 

EXERCISE  125. 

To  Introduce  the  Coefficient  of  a  Surd  under  the 
Radical  Sign. 

325.  It  is  sometimes  convenient  to  have  a  surd  in  a 
form  without  a  coefficient.  The  coefficient  can  always 
be  introduced  under  the  radical  sign  by  reversing  the 
process  of  Art.  324. 

(1)  Thus,  2l/G=l/22]/6 

=  V2'XG  by  [1]. 

_=l/24_ 

(2)  Also,  501/50=  1/502  ]/50 

=  l/50-^xr)0  by  [1]. 

=  1/1 250  JO 

(3)  Also,  4l^5=f/4^|/5 

=  1/4^5  by[l], 

=  1^320 


3l8  SURDS. 

o26.  As  the  same  process  may  evidently  be  applied 
in  any  case,  we  say  : 

Any  co'-Jjficicjit  of  a  S2ird  fiiay  be  introduced  as  a  factor 
tinder  Ike  radical  sign,  pi ovided  the  coefficient  be  first 
raised  to  a  fewer  equal  to  the  index  of  ike  surd. 

Examples. 

Place  the  coefficient  of  the  surd  beneath  the  radical 

sign  in  each  of  the  following  : 

1.  2l/2.  8.  41/4.        15.  xt'\        22.   ■-2#^J. 

2.  51/7.  9.   2l''3.         16.  y-f/~Q,.         23.  aV~b. 

3.  61/5.  10.  3)'  3.        17.  2-1^7.        24.  w^i/J^. 

4.  2l/'21.  II.  101/4.      18.  (-<^)i/g  25.  \VZ. 

5.  3l>^2.  12.  9^10.      ig.  (-^)^7  26.  1^^. 

6.  7T>^3.  13.   81/4I.       20.  -;^l>  10   27.   fl^T?. 

7.  81^5.  14.  aVb.         21.  —hV a.    28.  |V18p. 

EXERCISE  126. 

To  Integralize  the  Expression  under  a  Radical  Sign. 

227.  The  expression   under  the  radical  sign  of  any 
surd  can  always  be  made  a  whole  number. 
(1)  Thus,  i/f=i/|^=l/j 

=  l/ix2 

=  F  iy2  by  [1]. 

=  ^1''2  because  V \=\. 


>(2)  Also,  ^'|=l^'|xt=#'|J 


=  f/-2^7-Xl8 

^^/J,f'Yz  by[l]. 

=if/18 


TO    INTEGRAUZK    A    SURD.  3 19 


(3)  Also,  i?/|=i'-'|x|=rij 


=  1^3^x14 

=  rT>8#14  by[l]. 


(4)  Also.  V|=Vf x^=V^ 


=v 


1 

b 


n 


=.^-t-J/ab"-\  by[l]. 


^^i^ab"-' 
0 

328.  As  the  same  process  can  evidently  be  applied  in 
any  case,  we  may  say  : 

The  expression  under  the  radical  sign  in  any  surd  can 
be  made  integral  by  vinltiplying  both  numerator  ajid 
denominator  by  such  a  number  as  tvill  render  the  dowm- 
iiiator  a  perfect  power  of  the  indicated  root,  and  theti  taking 
the  root  of  tJie  denominator  thus  found. 
Examples. 

Integralize  the  expression    under  the  radical  sign  in 
each  of  the  following  : 


I. 

I'^i- 

6. 

1/*. 

II. 

1/1 

16. 

151/^. 

2. 

l/i. 

7. 

V-i,. 

12. 

\yj. 

17- 

fJ^K- 

3. 

v\ 

8. 

Vi\. 

13- 

^'h 

i3. 

4. 

v'i. 

9. 

rl 

14. 

ru^ 

19- 

l/2f. 

5. 

Fa 

10. 

^'l 

15- 

l/i0  8_ 

20. 

tl/f 

«\  r' 

11. 

Vr 

23- 

25- 

27. 

!2. 

3/^ 

24. 

2\  x^ 

26. 

1  ab^ 

\2x^- 

28. 

2.3/  3x 

320  •  SURDS. 

EXERCISE  127. 

To  Lower  or  Raise  the  Index  of  a  Surd. 

329.  We  can  change   the   index  of  any  surd  in  the 
following  manner : 


(1)  Thus,  i/4=^  1/4  by  [3] 

=  l/2  since  ]/4=2 


(2)  Also,         1^1000=  1^  1^1000  by  [3] 

=  l/ 10  since  1^  1000=  10 

(3)  Also,  1^2oG^2;t8  =  l'^  l/25(k^  by  [3] 

330.   Since  equation  [3]  is  true  in  all  cases,  we  know 
T/ie  index  of  a  siwd  can  be  lowered  if  the  expression 

under  the  radical  sign  is  a  perfect  power  corresponding  to 

some  factor  of  tlie  original  radical  index. 

^  Examples. 

I^ow^er  the  index  of  each  of  the  followinT  surds  : 


1.  173G.        6.   1^1000.    II.    t/a-b\     i6.   fU^x-'y^. 

2.  1^^2500.    7.  '1^2502.    12.  T255.       17.   1^'J. 

3.  I^IG.        0.  'i/2882.     13.  '1VT28.      18.  riJf. 


4.  FIG.        9.   1^27.        14.  F  8000.    19.  3l'/4y«2/>«^8 

5.  f^'m^  10.  1^27^.    15.  Vis\.      20.  4|^J^^^^. 

EXERCISE  128. 

To  Reduce  a  Surd  to  its  Simplest  Form. 

331.  A  surd  is  in  its  Simplest  Form  when,  (1)  no 
factor  of  the  expression  under  the  radical  sign  is  a  per- 
fect power  of  the  required  root ;  (2)  the  expression  under 
the  radical  sign  is  integral ;  (3)  the  index  of  the  surd  is 
the  lowest  possible. 


TO    REDUCE    TO    SIMPLEST   FORM. 


321 


332.  Methods  of  making  the  different  reductions  re- 
quired by  this  definition  have  already  been  explained  in 
exercises  124.  126,  and  127.     We  give  a  few  examples. 


(1)  Simplify 


^^'  V^' 


=  2-/4.^ 


(2)  Simplify  ^'/^^-. 


=il/60 
(3)  Simplify  |l^fii_ 

=  5VJ 
=  1/10 


by  exercise  127. 
by  exercise  126. 

by  exercise  127. 
by  exercise  126. 
by  exercise  124. 

by  exercise  127. 
by  exercise  124. 
by  exercise  126. 


333.  In  any  piece  of  work  it  is  usually  expected  that 
all  the  surds  will  finally  be  left  in  their  simplest  form. 


Examples. 
Reduce  each  of  the  following  surds  to  its  simplest  form: 

I.    V  ^.  ^.    V  ^.  5.    I    ^f.  7. 

6.    r  A.  8. 


2.  l^'V-. 


3.  i^¥ 

4.  f/u 


2T- 


JlL. 

V 


EXERCISE  129. 

Addition  and  Subtraction  of  Surds. 
334.  Surds  which  differ  only  in  their  coefficients  are 
said  to  be  Similar.     Thus,  61^2  and  15l/2  are  similar 
surds  ;  also  fl^f  and  }/^y ;  also  of  ab-  and  nfab"^. 


322  SURDS. 

335.  The  addition  and  subtraction  of  surds  involves 
no  principle  not  already  used  in  the  addition  and  sub- 
traction of  other  expressions,  as  the  following  examples 
show:  _  _ 

(1)  Combine  the  terms  of  10 1/?- 31/ 74- 51/7. 

101/7-31/7  +  51/7=121/7, 
by  the  usual  process  of  addition  of  terms. 

(2)  Combine  the  terms  of  7K  2  — l/l8  +  2l/8. 
Putting  each  surd  in  its  simplest  form,  we  have 

7l/2_i/l8+2l/8=7l/2-3l/2+4l/2 
=81/2 
by  the  usual  process  of  addition  of  terms. 

(3)  Combine  the  terms  of  5l/4  +  2l/32-l/l08. 
Putting  each  surd  in  its  simplest  form,  we  have 

6l/4  +  2l/32-l/l08=5l/4+4l/4-3l/J 
=61/4 

(4)  Combine  the  terms  of  |l/|-|l/f. 
Putting  each  surd  in  its  simplest  form,  we  have 

|i/|-i>^f=ii^6-ii/6 


=11/6 


(5)  Combine  terms  of  22f'a'd-Sa''l^QU  +  haf'aH. 

Putting  each  surd  in  its  simplest  form,  we  have 
22l/^-3a2i/64^+5«l/^ 
=22«2i/A-12^2^^^5«2^? 

=  15«2#/^ 

336.  We  observe  the  advantage  of  reducing  each  of 
the  surds  in  any  given  expression  to  its  simplest  form, 
for  then  it  can  be  told  whether  or  not  some  of  the  surd 
are  similar  to  each  other,  and  consequently  whether  or 
not  they  can  be  combined  together;  for  only  similar  surds 
can  be  combijied  into  a  single  surd. 


MULTIPLICATION    AND    DIVISION.  323 

Examples. 

Express  each  of  the  following  expressions  in  as  few 
terms  as  possible  : 

1.  21/3  +  31/3.  6.  2f/32-l/l08. 

2.  71/2-3 1/2.  7.  41/2-]^"  64. 

3.  11i/13-4t/13.  8.  1^243-51/48. 

4.  71/2  +  1/18.  9.  1/^-1/20. 

5.  4l/4-3l/4+2#^4.  10.  il/245  +  5l/i. 

11.  i^'':t:._-,^^_3,^256+ 1/625. 

12.  2v/|+l/60-l/l5  +  l/S. 

13.  2+Jl/|-|>/f.        _ 

'       15.   l/«^  +  2l/a— 1/4«. 
16.  l/^+il^V-3l/27a2. 


17.  l/27^*-l/8^*  +  |/l25f. 


18.    l^a*x+Vd\r-]^4aH''x. 


19.  6«l/63^^='-3l/ll2«^^/^3 +5^1/28^33. 

EXERCISE  130. 

Multiplication  and  Division  of  Surds. 

337.  The  product  of  any  number  of  surds  of  the  samf 
index  can  always  be  expressed  as  a  single  surd  by  means 
of  equation  [1].  _         _        _ 

(1)  Find  the  product  of  l/2  x  l/5  x  Vl. 

V^2 X  1/5 X  1/7  =  l/2x5x7  by  [1]. 

=  1/70 


324  SURDS. 

(2)  Find  the  product  of  l/2  X  "/I8. 


|/'2xl/ 18=1/2x18  by[l]. 

=6 
(3)  Find  the  product  of  V^/54x  1^9. 

l5^54xi/9=l/54x9  by  [1]. 

=  31^/2 
The  result  should  always  appear  in  its  simplest  form. 

338.  The  quotient  of  two  surds  of  the  same  index  may 
be  expressed  by  means  of  equation  [2]. 

(1)  Find  the  quotient  of  |/28-^|/7. 

V'28-^1/7=1/V-  by  p]- 

(2)  Find  the  quotient  of  1^81 -7- l^G. 

^/81^f/6=r-V--r^-  by  [2] 

=  ?)f\  by  exercise  124. 

=#1/4  by  exercise  126. 

The  result  should  always  appear  in  its  simplest  form. 

339.  If  the  product  or  quotient  of  surds  of  different 
indices  is  sought,  the  surds  may  first  be  reduced  to  a 
common  index  by  exercise  127. 

(1)  Find  the  product  of  v'5  X  P'4. 

V  '5  X  #4=  1'^  125  X  1/  IB      by  exercise  127. 
=  ri25xTG=V  2000  by[l]. 

(2)  Find  the  quotient  of  ^9-^V  3. 

f  9  _^  1    3  =  1'  81  ^  V  27        by  exercise  127 . 
==f'|l  =  ,V3  by  [2]. 


MULTIPLICATION    AND    DIVISION.  325 

(3)  Find  the  product  of  v' ad'^  xi  a'-b. 

|/^X  t\iH=  "Va'^b^-^  X  'f  V7-    by  Ex.  127. 
=  'f/^«^  by[l]. 

=  b  ^Va^b'^  by  exercise  124. 

(4)  Find  the  quotient  of  f^'^x^'V'^. 

>^  -K2nV^=rM-^2Y/;^         by  Kx.  127. 
=  i'>^i|x^f^  by  [2]. 


=4Y  16xlG=^#'l6  bv  Ex.  127 


Examples. 

Perform  the  indicated  operations  in  each  of  the  following 
and  leave  the  results  in  their  simplest  form  : 

1.  1/5x1/20.     18.  1/50-T-1/6.      35.  ^y^xf'^. 

2.  f'Zxf'n.  19.  l/96H-iV^3.      36.  bV^xt'oH^. 

3.  l^6x#^32.  20.  #'8i-^#'6.       37.  af/Jaxf^^K 

4.  l/2xl/'l2.  21.  l/80--1^20.     38.  #'^^xl^>7^* 

5.  fAxf^.  22.  y  8F--1/2F  39.  3l/27x5v''2?^. 

6.  l'/6xl'/8.  23.  #'547^^1'/ 2^  40.  #9^1/3. 

7.  ^^'4x1^10.  24.-  v'|H-]/f.         41.  #^72h-i/6. 

8.  f/81x#^45.  25.  V^if-^l/^.      42.  f'l2-M/6. 

9.  f  7x1'/^^^  26.  V'^V'i._      43-  l^'l^-l^'^f. 
10.  #'50xf/40.  27.  i/ff^i/^.     44.  1^/24^61^3. 

II.  I'/iix  1^121  28.  i/5xf/4._   45.  i/^-^rej. 

12.  2l/«xl/9^  29.   1/125  xi/36.   46.  5h-i/3. 

13.  aVyxbVy^   30.   l^|xi/f.         47-   30-- 1/210. 

14.  1/157x1/5731.   l/fxi/^.         48.   l/^H-l/'^. 

15.  2i/^x3f/^32.  i/eixi^Te.    49.  i/2;^-^v^. 

16.    i'48--l"3.      33.   #''l2x'lV97).    50.   #y--|/2^. 


17- 


45-- 1    10.    34.   f'fxl'^f         51.    jV^5^«-T-l/(^M 


326  SURDS. 

52.  (3  +  2l/5)C2-l/5). 

53.  (8+3l/2)(2-l/2). 

54.  (5+2i/3)(3-5t/3). 

55.  (3-l/6)(6-3l/6). 


56.  (i/5-t/9)(v'5+i/9). 

57.  (1/3 -1/2)  (1/3 +1/2). 

58.  (1/9  _v/r7)  (1/94.1/17), 

59.  (]/6+i/1i)(i^6-t/1i). 

60.  (T/9-|,v/lf)(i^9_v/i7), 

61.  (^12 +1/19)  (l^'l2 -1/19). 

EXERCISE  131. 

Powers  and  Roots  of  Surds. 

340.  Any  power  of  a  surd  can  be  expressed  as  a  single 
surd  by  means  of  the  principle  of  Art.  323.     Thus  : 

(1)  Square  ^2.  _  _ 

(l/2)2  =  #^22  by  [5]. 

=  ]/4 

(2)  Cube  31/2. 

(3l/2)3=33(i/2)'   by  law  Z>  for  indices. 

=27l/'2^  by  [5]. 

=  541/2  by  exercise  124. 

The  result  should  always  be  left  in  its  simplest  form. 

341.  Any  root  of  a  surd  can  be  expressed  as  a  single 
surd  by  means  of  the  principle  of  Art.  321.     Thus  : 

(1)  Find  square  root  of  l/4. 

y'~Ft=^yz=r';7t  by  [3]. 

=1/2 


POWERS    AND    ROOTS.  327 

(2)  Find  cube  root  of  |l/3. 

f'sy'S==2^iVs  by  exercise  124. 

=2^1/^=21^1/^  by  Ex.  125. 

-=2^f^  by  [3]. 

=2l/|=|l/3        by  exercise  126. 

(3)  Find  the  /th  root  of  aVJ. 

l/ai/b^^  V~arb  by  exercise  125. 

=  f/^  by  [3]. 

342.  This  last  process  is  a  general  one,  but  if  for  any 
particular  values  of  a,  b,  r,  and  t  this  result  should  not 
happen  to  be  in  its  simplest  form,  it  should  be  so  reduced. 

Examples. 
Express  each  of  the  following  as  a  surd  in  simplest  form: 


I. 

(Vby. 

7.  (ii^«^)^ 

13.  "^Wh 

2. 

(fzy. 

8.   {af'iY. 

14. 1^  1/16. 

3- 

(2l/2)8. 

9.   1^21/8. 

15.   I^3l/9«i«. 

4. 

(^^2)2. 

10.   1^^^36. 

16.  l/l28 1/243^7 0 

5. 

(1^)^ 

II.  ^31/3. 

17.     l/_l_^2|K9^2^2 

6. 

(-1/8)^ 

.     12.  ^vi 

18.  l^|l/-^. 

19. 

2"^  i/a-4.'^a  +  2^  f'a^} 

EXERCISE  132. 

Rationalization  of  Expressions  Containing 
Quadratic  Surds. 

343.  To  Rationalize  an  expression  is  to  perform  an 
operation  upon  it  that  will  free  the  expression  of  surds. 
Thus,  the  binomial  quadratic  surd  3  +  1/  2  is  rationalized 
when  multiplied  by  3  — 1/2,  for  the  product  is  9—2  or  7, 
which  is  rational. 


328  SURDS. 

344.  Any  multiplier  which,  when  applied  to  an  irra- 
tional expression,  will  free  the  expression  from  surds,  is 
called  a  Rationalizing  Factor.  Thus,  S~V2  is  a 
rationalizing  factor  for  the  binomial  surd  3-^-1^2. 

345.  It  is  often  convenient  to  perform  an  operation 
upon  both  terms  of  a  fraction  so  as  to  render  either  the 
numerator  or  the  denominator  rational.  It  is  suflScient 
for  present  purposes  to  show  how  this  may  be  done  when 
all  the  surds  are  of  the  second  order.  The  following  are 
examples. 

2 

(1)  Rationalize  the  denominator  of j^.    . 

2+y2 

Multiplying  numerator  and  denominator  by  2  — V^2, 
we  get  _  _ 

2 2(2-l/2)        _   4-2y2 

2-m/2~(2  +  1^2)(2— ]/2)~"4-(l/2)2 

Q 

(2)  Rationalize  the  denominator  of 


t/5-V'^2"_ 

Multiplying  numerator  and  denominator  by  Vb-\-\^2, 
we  get  _        _ 

3        ^  3(t/5  +  i/2) 

l/5-l/2  ~(|/5-t/2  )(t/54- l/2  ) 

^8(1/54- 1/2)^^-_^^- 
o — z 

346.  This  work  is  based  on  the  very  evident  principle 
that  any  bmomial  quadratic  surd  is  made  rational  by 
multiplying  it  by  itself  with  the  sign  of  07ie  of  its  terms 
changed,  for  the  product  is  the7i  the  difference  of  tivo  squares. 


RATIONALIZATION    OF    EXPRESSIONS.  329 

347.  Considerable  labor  is  often  saved  in  computing 
the  value  of  an  irreducible  fraction  if  we  first  rationalize 
the  denominator.     Thus,  to  compute  the  value  of 

3 
1/5-1/2 
to  five  decimal  places,  the  two  square  roots  must  be 
taken  to  at  least  five  decimal  places,  and  the  quotient  of?* 
divided  by  the  differe7ice  of  these  roots  must  be  foimd.  This 
division  by  a  five-place  number  will  be  avoided  if  we 
first  rationalize  the  denominator,  for  the  result  isl/54-  V^2, 
the  value  of  which  is  found  without  the  long  division  of 
the  former  method. 

Examples. 

Rationalize  the  denominator  in  each  of  the  following  : 

11  2v/2 

"  ■  -'^'  ^'  1/8-1/7'  ^*  81/2 -2V^3' 

(5  211/3 

10. 


Vb  V' 12  + 1/5  4V  8-31/2 

9  2  1/5-1/3 

7.    -7^—7zz'  II-    -7^ 


^-  5-l/'l0                  1/3  +  1-^2  *  1/5  +  1/3 

4  o                 3  ^^ 

4.   ^-  8.   —7= --•       12.   -j^ j== 

6-21/3                   V 10 -1/6  l/3  +  V'^2 


13- 


i-v'~i+v'^ 


1  +  1/2-1/3 

-N2+V^        (l-V'2-i-V3)  {l-f-V'2-fV'3)        1  +  2^3  +  3-2     V2+V6 


l-fV2-V3     {[1+V2]-V3)  ([H-V2J+V3)     1  +  2V2  +  2-3  2 

30 2+1/6-1/2 

^^'    2-1/3  +  1/5  ^^"    2-1/6+1/2 

16.    Find,  in  the  shortest  wav,    the  value  of 


to  four  places  ;  given  1    2  =  1.4142  + 


3  +  21/2 


350  SURDS. 

EXERCISE  133. 

Rationalization  of  Equations. 

348.  If  the  unknown  number  in  an  equation  appears 
under  a  radical  sign,  the  equation  must  first  be  rational- 
ized before  the  value  of  the  unknown  number  can  be 
found.     This  is  illustrated  by  the  following  examples. 

(1)  Solve  1/5^=20. 

Squaring  both  sides,  we  get 

5x=400, 
whence  .;»;=80. 


(2)  Solve  SVix-S^VlSx-S. 
Squaring  both  sides,  we  get 

2(4:X-S)=lSx-3, 
Transposing  and  uniting, 

23;i;=69, 
whence  x=S. 

(3)  Solve  l/^T9=5l/J-3 
Squaring  both  sides,  we  get 

x  +  d=25x—S0Vx-\-d. 
Transposing  rational  terms  to  one  side  and  irrational 
terms  to  the  other,  we  get 

24;t=30l/^ 
Dividing  by  6  and  squaring, 

16;t:2  =  25.;»;. 
Solving  this  quadratic,        ^=14  o^  ^' 


(4)  Solve  VS2+x=16-Vx. 

Squaring  both  sides,  _ 

32-f:»r=256-82l/;^+.^. 
Transposing,  uniting,  and  dividing  by  3i, 

whence  .r=49. 


RATIONALIZATION    OF    EQUATIONS.  33  I 

Examples. 
Solve  each  of  the  following  equations  : 

1.  1^2^=4.  10.   l/x—Vx'— 5=1/5. 

2.  1/8^=2.  li.  l/x—7=v'x—U  +  l, 

3.  }/x+A=4.  12.   l/^'7=1  '-r+1— 2. 

4.  }/2x+Q=4:.  13.  .r=7  — 1   x-^— 7. 

5.  ]/iar+l6=5.  14.   V^^+20-l/.r^-3=(). 

/;r ;;-         /- ~  X — 1  V  X-\-l 

6.  V2x-^1=V  5x—2.      15.        -       = V- 

V  Jtr-1        X-^ 

7.  14+:^4.r-40=10.      16.  -?II^=?±L£. 

8.  }/lG^T9=4l/4r-3  17.   v/jt--f3+l/'3x— 2=7. 

9.  l/^f+^=f+V^.       18.   1/2^+1+1/^^= 2 1/^. 


CHAPTER  XVIII. 
RATIO,   PROPORTION   AND   VARIATION. 

EXERCISE  134 

Ratio. 

349.  The  relative  magnitude  of  two  numbers  or  quan- 
titie.s,  measured  by  the  number  of  times  that  the  first 
contains  the  second,  is  called  the  Ratio  of  the  two  num- 
bers or  quantities. 

Thus,  12  contains  3  just  4  times,  hence  the  ratio  of  12 
to  ')  is  4  or  ^f-.     And  similarly  if  a  and  b  stand  for  any 

two  numbers  the  ratio  oi  a  \.o  b  is  -r- 

0 

350.  We  may  speak  of  the  ratio  of  two  quayitities  of 
the  same  kind  as  well  as  the  ratio  of  two  7iumbers.  Thus, 
12  feet  contains  3  feet  just  4  times,  hence  the  ratio  of  12 
feet  to  3  feet  is  4  or  -^^. 

The  student  should  notice  that  the  ratio  of  12  feet  to  3  feet  is  -^, 

12  feet 
not  —7 ,  for  division  proper  cannot  be  performed  except  when  the 

divisor  is  a  number.  Division  implies  separating  into  a  certain  num- 
ber of  equal  parts.  For  example,  a  stick  12  feet  long  may  be  sawed 
into  3  equal  pieces,  and  one  of  these  pieces  is  4  feet  long  ;  so  12  feet 
divided  by  3  is  4  feet.  But  while  a  stick  12  feet  long  may  be  divided 
by  3  but  not  by  3  feet,  still  a  stick  12  feet  long  may  be  measured  by  a 
stick  3  feet  long.  This  measured  by  is  often  confused  with  division 
proper. 

The  ratio  of  12  feet  to  3  feet  is  not  12  feet  divided  by  3  feet,  but  it 
is  12  divided  by  3. 

The  ratio  of  any  two  quantities  of  the  same  kind  may  be  looked 
upon  as  the  number  of  units  in  the  first  divided  by  the  number  of 
units  in  the  second.     Plainly,  quantities  which  are  not  of  the  same 


RATIO.  333 

kind  cannot  have  any  ratio,  for  one  cannot  possibly  be  measured  by 
the  other,  nor  can  one  be  contained  at  all  in  the  other.  For  example, 
ten  miles  cannot  be  measured  by  two  quarts,  nor  can  two  quarts  be 
contained  any  number  of  times  in  ten  miles. 

361.  The  ratio  of  a  to  d  is  denoted  in  either  of  two 
ways :  Jij's^,  by  writing  the  a  before  the  d  with  a  colon 
between  them,  thus,  a  :  l>;  second,  by  a  fraction  in  which 

a  is  the  numerator  and  b  is  the  denominator,  thus,  -  • 

0 

Whichever  way  the  ratio  is  written,  it  is  read  ''the  ratio 
of  a  to  ^, "  or  simply  '  'a  to  b. ' ' 

352.  In  either  waj^  of  writing  the  ratio  of  a  to  b,  a  is 
called  the  Antecedent  or  First  Term,  and  b  is  called 
the  Consequent  or  Second  Term. 

353.  Since  a  ratio  is  the  quotient  obtained  by  dividing 
the  number  of  units  in  the  antecedent  by  the  number  of 
units  in  the  consequent,  it  follows  that  the  properties  of 
ratios  may  be  obtained  immediately  from  the  properties 
of  fractions. 

354.  Since  a  fraction  may  be  multiplied  either  by 
multiplying  the  numerator  or  dividing  the  denominator, 
it  follows  that  a  ratio  may  be  tnultiplied  either  by  tnulti- 
plying  the  antecedent  or  by  dividincr  the  conseipient. 

355.  Since  a  fraction  may  be  divided  either  by  divid- 
ing the  numerator  or  multiplying  the  denominator,  it 
follows  that  a  ratio  may  be  divided  either  by  dividing  the 
antecedent  or  by  multiplying  the  consequent. 

356.  vSince  a  fraction  remains  unchanged  in  value 
when  both  numerator  and  denominator  are  multiplied  or 


334  RATIO,    PROPORTION   AND    VARIATION. 

divided  by  the  same  number,  it  follows  that  a  ratio  temai7is 
unchanged  in  value  ivheii  both  antecedent  and  conseque7it  are 
multiplied  or  divided  by  the  same  number. 

357.  If  the  numerator  of  a  fraction  is  greater  than  the 
denominator,  the  fraction  is  greater  than  1  ;  therefore, 
if  the  antecedent,  of  a  ratio  is  greater  than  the  consequent, 
the  ratio  is  greater  tha7i  1. 

358.  If  the  numerator  of  a  fraction  is  less  than  the 
denominator,  the  fraction  is  less  than  1  ;  therefore,  if  the 
antecede7it  of  a  ratio  is  less  than  the  consequent^  the  ratio  is 
less  than  1. 

359.  If  the  numerator  and  denominator  of  a  fraction 
are  equal  to  each  other,  the  fraction  is  equal  to  1  ;  there- 
fore, if  the  antecedent  and  conscque?it  of  a  ratio  are  equal  to 
each  other ^  the  ratio  is  equal  to  1. 

360.  Theorem.  A  ratio  which  is  greater  than  1  is  de- 
creased by  increasing  both  antecedent  and  consequent  by  the 
same  amoimt. 

Let  T  be  a  ratio  which  is  greater  than  1  ;  then  a'^b. 

Now  form  a  new  ratio  by  increasing  the  antecedent  and 
consequent  by  the  same  amount,  x.     The  new  ratio  is 

a-\-x 
'bTx' 
If  we  multiply  antecedent  and  consequent  of  the  original 
ratio  by  d-{-x,  we  get 

a_ab-}-ax 

'b~  b-'+bx  ^^^ 

If  we  multiply  antecedent  and  consequent  of  the  new 
ratio  by  b,  we  get 

a-{-x_ab-i-bx 

b-]-x~b-  +  bx  ^^ 


RATIO.  335 

Now,    as   a><^,    it   is   plain   that   ax^bx,    and   hence 
a.b-{-ax'>ab-\-bx.     Therefore, 

ab-\-ax     ab-\-bx 

'WVbx-^  b'^^bx 
Therefore,  from  (1)  and  (2), 

a     a-\-x 

ly^T^x 

which  is  what  was  to  be  proved. 

->1 


Since  a^b,  it  is  plain  that  a^x^b-\-x.     Therefore 
b+x' 


Therefore,  as  each  ratio  -y  and  -r- —  is  greater  than  1 ,  and 
o  o-j-x 

as  -r'>T-. — ,  it  follows  that  y- —  ts  nearer  the  value   1 
b     b-j-x  b+x 

a 
than  —  is, 

0 

361.  Theorem.  A  ratio  which  is  less  than  1  is  in- 
creased by  increasing  both  ajitecedent  and  consequent  by  tlie 
same  amo2int. 

Let  T  be  a  ratio  which  is  less  than  1;  then  a<^b.    Now 
o 

form  a  new  ratio  by  increasing  the  antecedent  and  conse- 
quent by  the  same  amount,  x.     The  new  ratio  is 

a-\-x 

b+x' 
If  we  multiply  antecedent  and  consequent  of  the  original 

ratio  by  b-\-x,  we  get 

a_ab+ax  .^. 

'b~b^-{-bx'  ^^ 

*  This  notation  denotes  that  a  is  less  than  b.  Whenever  this  symbol  is  used 
the  point  of  the  angle  is  toward  the  lesser  and  the  opening  toward  the  greater 
number.     See  note,  page  291. 


336  RATIO,    PROPORTION   AND    VARIATION. 

If  we  multiply  antecedent  and  consequent  of  the  new 
ratio  by  b,  we  get 

a-[-x^ab-\-bx 

'b+x     b'^^bx  ^^ 

Now,  as  a<^b,  it  is  plain  that  ax<^bx^  and  hence 
ab-\-ax<^ab-{-bx.     Therefore, 

ab-\-ax  ^ab-\-bx 
W+bx^'PTbx' 
Therefore,  from  (1)  and  (2), 

a  ^a-\-x 
l^'b+i' 
which  is  what  was  to  be  proved. 

Since  a<,b,  it  is  plain  that  a+x-Cb  +  x.     Therefore, 

b+x 

Therefore,  as  each  ratio  -r  and  -r-i —  is  less  than  1,  and  as 

0  o-\-x 

a     a-\-x  .    ^  ,,  ,       a  +  x  .  ,,         ,      ^   ,,       ^  . 

-7-<-i— — ,  it  follows  that  ,— —  IS  nearer  ilie  value  1  than  ,-  is. 
b     b+x  b+x  b 

This  last  statement,  together  with  the  last  statement 
in  the  previous  article,  shows  that  any  ratio  (except  the 
ratio  1)  is  made  more  nearly  the  value  1  by  i7icreasing  both 
antecedent  ayid  conseqtient  by  the  same  a^nount. 

362.  If  from  two  given  ratios,  ,-  and  -or  a\b  and  c :  d, 

b  d 

we  form  another  ratio  by  multiplying  the  antecedents 
together  for  a  new  antecedent  and  the  consequents  to- 
gether for  a  new  consequent,  we  get  t;^  or  ac\  bd,  which 
is  said  to  be  Compounded  of  the  given  ratios  a  :  b 
and  c  :  d. 


RATIO.  337 

363.  If  ^>1  it  follows  that  xy^j/,  or,  expressed  in 
words,  if  the  multiplier  is  greater  than  1  the  product  is 
greater  than  the  multiplicand. 

Therefore,  if  :i>  1,  t  ,i>t-     Also,  if  ->  1,  ^  -^>4- 
a  0  a     0  0          odd 

But  in  each  of  these  cases  7-  -  or  — ,  is  the  ratio  com- 

0  d       cd 

d  c 

pounded  of  the  ratios  —  and  -j- 

Therefore,  if  a  ratio  be  compounded  of  tivo  ratios  each  of 
which  is  greater  tha7i  1  the  result  is  greater  than  either  of 
the  given  ratios. 

364.  In  a  similar  way  it  may  be  .shown  that  if  a  ratio 
be  compoimded  of  two  ratios  each  of  which  is  less  than  1  the 
result  is  less  than  either  of  the  given  ratios. 

365.  If  x<y  it  follows  that  xy<Cy,  or,  expressed  in 
words,  if  the  multiplier  is  less  than  1  the  product  is  less 
than  the  multiplicand. 

Therefore.  \i—<A,  t^<t,  and  it  was  before  shown 
d         0  d     0 

thatif^>l,^^>^. 

Therefore,  if  a  ratio  be  compoiinded  of  two  given  ratios, 
one  of  ivhich  is  greater  than  1  and  the  other  less  than  1,  the 
result  is  iiitermediate  in  value  betiveen  the  tivo  given  ratios. 

Problems. 

1.  Arrange  the  ratios  2:3,  3:5,  5:8  in  the  order  of 
magnitude. 

2.  Which  is  nearer  unity  2  :  3  or  8  :  9  ? 

3.  Which  is  nearer  unity  2  :  3  or  2-\-x  :  3+jtr? 

4.  For  what  value  of  x  will  the  ratio  84--^  :  36+^  be 

equal  to  the  ratio  1:3? 

22 


338  RATIO,    PROPORTION   AND    VARIATION. 

5.  What  must  be  added  to  each  term  of  the  ratio  9  :  16 
to  produce  the  ratio  3:4? 

6.  What  must  be  subtracted  from  each  term  of  the 
ratio  3  :  5  to  produce  the  ratio  5:9? 

7.  A  certain  ratio  will  become  equal  to  -J-  when  1  is 
subtracted  from  each  of  its  terms,  and  equal  to  f  when  9 
is  added  to  each  of  its  terms.    Find  the  ratio. 

8.  Find  two  numbers  such  that  if  each  is  increased  by 
1  the  results  have  the  ratio  2  :  3,  and  if  each  is  increased 
by  7  the  results  have  the  ratio  5:6. 

9.  Find  two  numbers  such  that  if  the  first  is  increased 
by  2  and  the  second  decreased  by  2  the  results  have  the 
ratio  6:5,  but  if  the  first  is  decreased  by  2  and  the 
second  increased  by  2  the  results  have  the  ratio  4:7. 

10.  A  rectangle  is  39  feet  long  and  36  feet  wide.  Ex- 
press the  ratio  of  the  length  to  the  breadth. 

11.  What  is  the  ratio  of  12  lbs.  8  oz.  to  21  lbs.  14  oz.? 

12.  What  is  the  ratio  of  5  ft.  to  6  ft.  3  in.? 

13.  Two  rectangular  fields  have  the  same  area.  The 
length  of  the  first  is  180  feet  and  of  the  second  150  feet. 
What  is  the  ratio  of  their  widths  ? 

14.  The  areas  of  two  rectangular  rooms  have  the  ratio 
2:3.  The  length  of  the  first  is  12  feet,  and  of  the  second 
20  feet.     What  is  the  ratio  of  their  widths  ? 

15.  Two  numbers  are  in  the  ratio  of  3  : 4,  and  the  sum 
of  their  squares  is  400.     What  are  the  numbers  ? 

16.  Two  equal  sums  of  money  are  on  interest ;  the 
first  runs  8  months  at  7  per  cent.,  the  second  runs  9 
months  at  6  per  cent.  The  interest  in  the  first  case  has 
what  ratio  to  the  interest  in  the  second  case  ? 


PROPORTION.  339 

17.  Divide  the  number  10  into  two  such  parts  that  the 
squares  of  these  parts  shall  have  the  ratio  9  : 4. 

18.  A  train  of  cars  travels  140  miles  in  3|  hours,  and 
another  train  travels  240  miles  in  5  hours.  What  is  the 
ratio  of  their  rates  of  speed  ? 

19.  Show  that  the  ratio  aid  is  equal  to  (a+x)-:  (d+xy 
\i  x'^  =  ab. 

20.  V/hat  must  be  the  value  of  x  in  order  that  the 
ratio  6  :  x  may  equal  b—x  ? 

EXERCISE  135. 

Proportion. 

366.  The  expression  of  equality  which  exists  between 
two  ratios  is  called  a  Proportion.  Thus,  a\b=c\d\S2i 
proportion.  In  this  case  the  four  numbers  or  quantities 
are  said  to  be  in  proportion. 

367.  Sometimes  the  proportion  is  expressed  as  an 
ordinary  equation  in  fractions,  thus,  -T—-7,  and  some- 
times by  the  notation  a:b::c:d.  However  written,  the 
proportion  is  read,  '  'a  is  to  ^  as  <:  is  to  ^. " 

368.  In  the  proportion  a:b=c:d  the  letters  a,  b,  c, 
and  d  are  called  the  Terms  of  the  proportion.  The  first 
and  fourth  terms  are  called  the  Extremes,  and  the 
second  and  third  terms  are  called  the  Means. 

369.  VL  a,  by  c,  and  d  are  proportional,  we  have  the 
proportion  a  :  b  ::  c:d,  or,  written  in  the  fractional  form, 

CL        C 

T=~j-     Now,  multiplying  each  member  by  bd,  we  get 

ad^=  be. 
That  is,  the  product  of  the  means  equals  the  product  of  the 
extremes. 


340  RATIO,    PROPORTION    AND    VARIATION. 

370.  If  we  have  given  the  equation 

ad=  be, 

then  dividing  each  member  by  bd,  we  get 
ad     be       a      c  .  . 

Hence,  if  the  produet  of  two  numbers  equals  the  produet  of 
two  other  numbers,  the  numbers  are  in  proportion,  or,  in 
other  words,  if  the  produet  of  two  numbers  equals  the 
produet  of  two  other  numbers,  the  factors  of  one  produet  may 
be  taken  for  the  extre7?ies  and  the  faetors  of  the  other  product 
may  be  taken  for  the  means  of  a  proportion. 

371.  From  the  equation  ad=bc  we  may  infer  either 

a\b=c\d  (1) 

or  '  a  :  c  =  b :  d  (2) 

or  b:a=d:c  ['-)) 

or  b:d=a\c  (4) 

and  therefore  from  (1)  we  may  infer  either  (2)  or  (3)  or  (4). 
The  proportion  (2)  is  said  to  be  deduced  from  (1)  1)y 
Alternation.  The  proportion  (3)  is  deduced  from  {1} 
by  Inversion.  The  proportion  (4)  is  really  the  same  as 
(2)  except  that  the  two  members  have  changed  places  ; 
it  may  be  deduced  from  (3)  by  alternation. 

372.  If  ^  :  b=c :  d,  then  writing  in  fractional  form, 

a      e 

Adding  1  to  each  member, 

a\-b     e+d 
or  in  another  form,       —7—= — v-, 
0  a 

or  in  the  common  form  of  proportion, 
a-{-b\  b=e-{-d:  d. 
This  last  proportion,   a-\-b  :  b=c-\-d \d,  is  said  to  be 
derived  from  the  proportion  a  :  b=c\  d  by  Composition. 


PROPORTION.  341 

d      c 

373.  If  «  :  b—c\  d,  then  t=;j,  and  subtracting  1  from 

each  member, 

b  d       ' 

or  in  another  form, 

a — b     c—d 

or  in  the  common  form  of  proportion, 
a  —  b:  b=c — d:  d. 
This  last  proportion,   a  —  b:b=c—d\d,  is  said  to  be 
derived  from  the  proportion  a  \b=c:d  by  Division. 

374.  If  a  :  b=c :  d,  then  by  composition, 

a-^b_c-\-d 
b  d   '  ^^^ 

and  by  division,  — ,    = — j  .  (2) 

p  d 

Divide  (1)  and  (2)  member  by  member  and  we  get 
a-\-b  _c-\-d 
a—b     c—d 
or  written  in  the  ordinary  form  of  a  proportion, 
a-\-b  ■.a—b=c-\-d:  c—d. 
This  last  proportion,  a-\-b  :a—b—c-\-d  \c—d,  is  said  to 
be  derived   from   the  proportion   a\b=c\d  by  Compo- 
sition and  Division. 

375.  The  products  of  conespondmg  terms  of  two  or  more 
sets  of  proportional  numbers  are  proportio7iaI. 

Let  a  :  b=  c  :  d 

Mild  e\f=h\k 

and  n\r=^  s:t 


342  RATIO,    PROPORTION   AND    VARIATION. 

Writing  each  of  these  proportions  as  an  equation  in 
fractions,  we  have 

a_  c 

f-k 
n     s 

and  from  these  equations  by  multiplication  we  obtain 

a  e  71  _  c  h  s 

l}fr~d'kt 
ae?i      chs 
^^  Ifr^'dki 

or  writing  this  in  the  ordinary  notation  of  proportion, 
we  have  aeyi  :  bfr—chs  :  dkt, 

which  is  what  was  to  be  proved. 

376.  Like  powers  or  like  roots  of  proportional  numbers 
are  proportional. 

Let  a  :  b=^c  :  d. 

Writing  this  proportion  as  an  equation  in  fractions,  we 

obtain  |=|,  (1) 

and  raising  each  member  to  the  7i  th  power,  we  have 

a''     ^" 

or  writing  this  in  the  ordinary  notation  of  proportion, 

we  have  a''\  b"=c'':d''.  (3) 

Again,  taking  the  nih.  root  of  each  number  of  (1),  we 

a^'      ^" 
have  -=_  .  (4) 

b-     d- 

or  writing  in  the  ordinary  notation  of  proportion,  we  have 

1-1       11 

Equations  (3)  and  (5)  are  those  which  were  to  be  proved. 


PROPORTION.  343 

377.  In  the  proportion  a  :  b=b  :c,  b  is  called  a  Mean 
Proportional  between  a  and  c,  and  c  is  called  a  Third 
Proportional  to  a  and  b. 

If  in  this  proportion  we  write  the  product  of  the  means 
equal  to  the  product  of  the  extremes,  we  get 

b'-^ab, 
or  extracting  the  square  root  of  each  member,  we  get 
b^V'^b, 
Therefore,  a  mean  proportional  between  two  numbers  is 
equal  to  the  square  root  of  their  product. 

378.  A  Continued  Proportion  is  a  series  of  equal 
ratios.     Thus,         a  :  b=c :  d=e  :/,  etc. 

379.  In  any  continued  proportion  any  a7itecedent  is  to 
its  corresponding  consequent  as  the  smn  of  all  the  antecedents 
is  to  the  sum  of  all  the  consequetits . 

Let  the  continued  proportion  be  written  as  several 
equal  fractions,  thus: 

ace 

IT'd^J 
and  let  r  be  the  common  value  of  each  of  these  fractions, 
then 

--f=r  or  a=^br, 

0 

— =r  or  c=dr, 
a 

— =r  or  e=fr. 

Therefore,  by  addition, 

a  +  c+e^br+dr^-fr^^iib+d^fy. 
Hence,  by  dividing  by  b-\rd-\-f, 

a-\-c-\-e _   _a 

b  +  d-[-f~'^~T 
which  is  what  was  to  be  proved. 


344  RATIO,    PROPORTION   AND    VARIATION. 

Examples  and  Problems. 

1.  Write  two  proportions  from  the  equation  5  x  8=4  x  10 

2.  Write  two  proportions  from  the  equation  ab—cd. 

3.  Write  two  proportions  from  the  equation  ab=ac. 

4.  Write  two  proportions  from  the  equation  x'^—yz. 

5.  Write  two  proportions  from  the  equation  (a-{-by  =  7ir 

6.  Write  two  proportions  from  the  equation  ?>n'^r=4:pq. 

7.  Form  a  proportion  with  the  numbers  3,  5,  21,  35. 

8.  Form  a  proportion  with  the  numbers  3,  20,  90,  600. 

9.  Form  a  proportion  with  the  numbers  3,  6,  20,  40. 
10.  What  must  x  stand  for  in  order  that  .r  :  3=6  :  9  may 

be  a  true  proportion  ? 

11.  What  is  the  value  of  x  \i  b  \x=%  :  12  ? 

12.  What  is  the  value  of  Jt:  if  12  :  lb=x  :  35  ? 

13.  What  is  the  value  of  .r  if  8  :  10=90  :  x  ? 

14.  What  is  the  value  of  ;r  if  jt: :  6=jr— 1  :  jir+ll  ? 

15.  What  is  the  value  of  x  if 

Ibia  +  b)    lQ{aby     a-b 


\^{a-b')'1\{aby     a^-b 
16.  What  is  the  value  of  x  if 


X? 


(^-)^(^+-)- 


17.  What  is  the  value  of  x  if 

x\  (5;z— 6r)  =  (3;z2-f  2;2r— 8r2)  :  {bn'^ ^A.nr—Vlr'^^  ? 

18.  What  is  the  value  of  x  if 

ia-^b)  :x={a''-\-ab^b'')  :  {a^—b^)  ? 

19.  If  a  :  b=c  :  d,  prove  a  :  inb-=c :  md. 

20.  If  a  :  b=c  :  d,  prove  na  :  rb=^nc  :  rd. 

21.  li  a  :  b=c\d,  prove 

{jia-\-rb)  :  {iia  —  rb)-={nc-\-rd^  :  (nc—rd). 


VARIATION.  345 

22.  If  a  :  b=c :  d,  prove  7ia  :  rb=~  :  — 

r    n 

23.  If  a  :  b^c :  d,  prove  that  a  :  a-\-c=ab  :  a-\-b-\-c-\-d. 

24.  If  a  :  b=c :  d,  prove  that 

a"^  -}-ab  +  b'-.  a-  —ab-^ b^=c'^  -}-cd-]-d^ ■  c"^  —cd+d'K 

25.  Prove  that  a  :  b=c:d,  if 

(a-^b+c+d)  (_a—b—c+d)  =  {a—b^-c—d')  {a-\-b—c—d^. 

EXERCISE  136. 

Variation. 

380.  Thus  far  whenever  a  letter  has  been  used  to 
stand  for  a  number,  it  has  always  represented  the  same 
number  throughout  the  same  problem;  or  in  other  words, 
the  value  of  the  letter  has  always  remained  the  same  in 
the  same  problem.  But  sometimes  it  is  necessary  to  con- 
sider quantities  which  change  in  value  in  the  same 
problem,  and  then  the  letter  used  in  connection  with  such 
a  quantity  may  stand  for  one  number  at  one  instant  and 
another  number  at  another  instant  in  the  same  problem. 
For  example,  if  a  train  of  cars  travels  at  the  rate  of  40 
miles  an  hour,  of  course  the  number  of  miles  traveled 
depends  upon  the  time  of  traveling.  Now  we  may  take 
a  letter,  say  x,  to  represent  the  number  of  miles  the  train 
travels,  and  another  letter,  say  jr,  to  represent  the  number 
of  hours  the  train  travels;  then  plainly  we  have 

x=iOy  or  —=40. 

Here  x  and  jj^  stand  for  numbers  which  are  ?zo^  the  same 
at  one  time  as  at  another;  or  in  other  words,  x  and  jy 
change  in  value,  but  however  much  or  little  they  change, 
the  ratio  of  these  remains  unchajiged,  or,  as  it  is  usually 
exprCvSsed,  the  ratio  reinains  constant. 


34^  RATIO,    PROPORTION   AND    VARIATION. 

381.  One  number  or  quantity  is  said  to  Vary  Directly 

as  another  when  the  number  of  units  in  the  first  is  equal 
to  the  number  of  units  in  the  second  multiplied  or 
divided  by  some  constant  number,  or  what  is  the  same 
thing,  when  the  ratio  of  the  number  of  units  in  the  first 
to  the  number  of  units  in  the  second  is  some  constant 
number. 

In  the  above  illustration  the  distance  traveled  is  said 
to  vary  directly  as  the  time  occupied,  or  the  number  of 
miles  traveled  varies  directly  as  the  number  of  hours 
occupied. 

382.  The  word  directly  is  often  omitted,  and  we  say 
simply,  that  one  number  or  quantity  varies  as  another. 
The  same  idea  is  often  expressed  by  saying  that  the 
second  number  or  quantity  is  Proportional  to  the  first 
number  or  quantity.  In  the  above  illustration  the  dis- 
tance traveled  is  proportional  to  the  time  occupied,  or  the 
number  of  miles  traveled  is  proportio?ial  to  the  number  of 
hours  occupied. 

383.  If  X  is  proportional  to  y  then  —=c  where  <:is  some 

y 

constant,  i.  e.  some  number  that  remains  unchanged. 
Now,  if  we  represent  some  particular  value  of  x  hy  x^ 
and  the  corresponding  particular  value  oi  y  by  j^,  then 

— ^=^  and  from  this  it  is  easily  seen  that  when  one  num- 
yx 

ber  is  proportional  to  another,  we  know  the  constant  by 
which  one  of  the  numbers  must  be  multiplied  to  produce 
the  other,  if  we  know  any  corresponding /^^r/z-f/z/^^r  values 
of  the  two  numbers.  For  example,  if  we  know  that  a 
traveler  is  walking  uniformily,  i.  e.  at  the  same  rate,  then 
we  know  that  the  number  of  miles  traveled  is  propor- 
tional  to   the   number  of  hours    occupied,  hence  if  we 


VARIATION.  347 

represent  the  number  of  miles  traveled  by  x  and  the 
number  of  hours  occupied  by  jk,  we  have 

X  i 

x^cy  or  — =^. 

y 

Now  if  we  further  know  that  the  traveler  walks  15  miles 

15 
in  5  hours,  we  have         <:=^=3. 

o 

From  this  we  see  that  3  may  be  written  in  place  of  r,  and 

X 

hence  we  know  x=Sy  or  — =3. 

384.  One  number  or  quantity  is  said  to  Vary  In- 
versely as  another  when  the  first  is  equal  to  some  con- 
stant divided  by  the  second.     Thus, 'Ji:  varies  inversely 

as  r,  if  X—  —  where  c  is  some  constant. 

y 

385.  If  in  the  equation  x=—  we  multiply  eac^j  number 

by  y,  we  get  xy=^c. 

Therefore,  we  say  that  one  number  varies  inversely  as  a 
second  when  the  prodtut  of  the  two  numbers  is  some  constaiit. 
For  example,  the  time  occupied  in  traveling  a  certain 
distance  varies  i7iversely  as  the  rate  of  speed,  or  the  number 
of  hours  occupied  in  traveling  a  certain  number  of  miles 
varies  inversely  as  the  number  of  miles  traveled  per  hour. 

386.  Instead  of  saying  that  one  number  varies  inversely 
as  another,  the  same  idea  is  often  expressed  by  saying  that 
one  number  is  Reciprocally  Proportional  to  another. 

387.  One  number  Varies  Jointly  as  two  others  when 
the  first  is  equal  to  some  constant   multiplied  by  the 

X 

product  of  the  other  two.     Thus,   if  x=cyz,   or — =<:, 

where  c  is  some  constant,  x  is  said  to  vary  jointly  as 
y  and  2,  The  same  idea  is  often  expressed  by  saying  that 
one  number  is  Proportional  to  the  Product  of  two  others. 


348  RATIO,    PROPORTION    AND    VARIATION. 

388.  One  number  is  said  to  vary  as  the  square  of 
another  when  the  first  is  some  constant  multiplied  by  the 
square  of  the  second,  or  when  the  first  divided  by  the 
square  of  the  second  is  some  constant.     Thus,  if  x=cy^, 

X 

or  — 2  =  ^j  where  c  is  some  constant,  x  is  said  to  vary  as  j^'^. 

The  same  idea  is  often  expressed  by  saying  that  one 
number  is  Proportional  to  the  Square  of  another. 

389.  One  number  is  said  to  Vary  Directly  as  a  second 
and  Inversely  as  a  third  when  the  first  is  equal  to  some 
constant  multiplied  by  the  ratio  of  the  second  to  the  third. 

Thus,  ii  x=c~ ,  x' \s  said  to  vary  directly  as  y  and  in- 

2 

versely  as  z. 

The  same  idea  is  often  expressed  by  saying  that  the 
first  number  is  Directly  Proportional  to  the  second 
and  Inversely  Proportional  to  the  third. 

Examples  and  Problems. 

1.  If  Ji;  is  proportional  to y,  and  x=45  when ^^=3,  find 
the  value  of  x  when  j>/=15. 

2.  If  Ji;  is  inversely  proportional  to  ^,  and  x=l  when 
y=6,  find  the  value  of  x  whenjK=15. 

3.  Form  a  proportion  with  the  two  sets  of  values  of  x 
and  y  in  problem  2. 

4.  If  X  varies  as  y^,  and  x=l  when  y=5,  find  the 
value  of  X  whenjj/=15. 

5.  Form  a  proportion  with  the  two  sets  of  values  of 
X  and  y^  in  problem  4. 

6.  If  X  varies  jointly  as  j/  and  z,  and  x=20  when  jj/=2 
and  2=3,  find  the  value  of  x  whenjK=3  and  2=6. 

7.  If  .r  varies  inversely  as  the  square  of  y,  and  x=l 
when  y  =10,  find  the  value  of  x  when  ^^=5, 


VARIATION.  349 

8.  If  ,r  is  directly  proportional  toj/  and  inversely  pro- 
portional to  z,  and  ;r=20  when  y=^  and  2'=  4,  j&nd  the 
value  of  X  whenjK=3  and  2'=  10. 

9.  If  X  varies  as^,  prove  that  .f  ^  varies  2s,  y^ . 

10.  If  X  is  inversely  proportional  to  y  and  y  is  inversely 
proportional  to  z,  prove  that  x  is  proportional  to  z. 

11.  If  ;t:  is  proportional  to  z  and^  is  also  proportional 
to  ?,  prove  that  xy  is  proportional  to  ^-'^  also  that  x'^+y'^ 
is  proportional  to  z"^ . 

12.  If  Sx+7y  is  proportional  to  Sx-\-VSy  and  x=r> 
whenjj'=3,  find  the  ratio  of  x  toy  and  thus  show  that  x 
varies  asy. 

13.  The  number  of  feet  a  body  falls  is  proportional  to 
the  square  of  the  number  of  seconds  occupied  in  falling. 
Knowing  that  a  body  falls  16  feet  the  first  second,  find 
how  many  feet  it  will  fall  in  5  seconds. 

14.  With  the  same  supposition  as  in  the  last  example, 
find  the  height  of  a  tower  from  which  a  stone  dropped 
from  the  summit,  reaches  the  ground  in  3|  seconds. 

15.  The  weight  of  a  metal  ball  is  proportional  to  the 
cube  of  the  radius,  and  a  ball  whose  radius  is  2  inches 
weighs  10  pounds,  what  is  the  weight  of  a  ball  whose 
radius  is  5  inches  ? 

16.  If  a  heavier  weight  draw  up  a  lighter  one  by 
means  of  a  cord  passed  over  a  fixed  wheel,  the  number  of 
feet  passed  over  by  each  weight  in  any  given  time  varies 
directly  as  the  difference  of  the  weights,  and  inversely  as 
the  sum  of  the  weights.  If  10  pounds  draw  up  6  pounds 
16  feet  in  2  seconds,  how  high  will  14  pounds  draw  10 
pounds  in  2  seconds  ? 


CHAPTER  XIX. 

PROGRESSIONS. 

EXERCISE  137. 

Arithmetical  Progressions. 

390.  An  Arithmetical  Progression  is  a  series  ol 
terms  such  that  each  term  differs  from  the  immediately; 
preceding  term  by  a  fixed  number,  called  the  Common 
Difference.  The  following  are  examples  of  arithmetical 
progressions  : 

(1)  2,  4,  6,  8,  10.  (3)  2i,  3f,  5,  6i,  7^ 

(2)  31,  26,  21,  16.  (4)  (x-y),  x,  (x-hy-). 

(5)  a,   (a-i-d),   (a-\-2d-),   (a  +  Sd). 

391.  The  first  and  last  terms  of  any  given  progression 
are  called  the  Extremes,  and  the  other  terms  are  called 
the  Means. 

1.  The  first  term  of  an  arithmetical  progression  is  3 
and  the  common  difference  is  2.  What  is  the  5th  term  ? 
What  is  the  10th  term  ? 

2.  The  first  term  of  an  arithmetical  progression  is  5 
and  the  common  difference  is  3.  What  is  the  4th  term  ? 
What  is  the  7th  term  ? 

3.  In  the  last  progression,  how  many  times  must  the 
common  difference  be  added  to  5  to  produce  the  4th 
term  ?  to  produce  the  7th  term  ?  to  produce  the  7i  th  term  ? 

4.  The  first  term  of  an  arithmetical  progression  is  19 
and  the  common  difference  —4.  What  is  the  third  term? 
What  is  the  fifth  term  ?     What  is  the  n  th  term  ? 


ARITHMETICAL    PROGRESSIONS.  35  1 

392.  The  n  th  Term  of  any  Arithmetical  Progres- 
sion. It  is  usual  to  represent  the  first  term  of  an 
arithmetical  progression  by  a,  the  common  difference  by 
d,  and  the  n  th  term  by  /.  With  this  notation  we  may 
represent  any  arithmetical  progression  by 

No.  of  term:     12  3  4  5    .   .   . 

Progression:     a,    (^  +  ^),    {a^-'W),    («  +  3^),    («  +  4^) 

We  notice  that  the  coefficient  of  d  in  the  2d  term  is  1 , 
in  the  od  term  is  2,  in  the  4th  term  is  3,  and,  by  the 
nature  of  the  progression,  the  coefficient  oi  d  in  any  term 
is  1  less  than  the  number  of  that  term.  Therefore,  the 
n  th  term  in  this  progression  will  be 

a-\-{7i—\)d, 
or,  representing  the  n  th  term  by  /,  we  have  the  formula 
l=a-^[n-\)d,  [1] 

393.  Evidently  the  sum  of  an  arithmetical  progression 
is  not  changed  if  the  order  of  the  terms  be  reversed;  thus, 

3  +  5  +  7  +  9  +  11  may  be  written  11  +  9+7  +  5  +  3 
2|-+3i+4  +  4f  may  be  written  4|+4+3^+2i 

in  which  case  the  first  term  becomes  the  last  term,  the 
last  term  becomes  the  first  term,  and  the  commoji  difference 
chmiges  sign. 

394.  When  the  common  difference  is  positive  the  pro- 
gression may  be  called  an  Increasing  Progression,  and 
when  the  common  difference  is  negative  the  progression 
may  be  called  a  Decreasing  Progression. 

395.  The  Sum  of  n  Terms  of  any  Arithmetical 
Progression.  If  s  stands  for  the  sum  of  71  terms  of  an 
arithmetical  progression,  evidently  we  may  write  the  two 


352  PROGRESSIONS. 

following  equalities,  the  progressions  being  alike  except 
written  in  reverse  order  : 

s=:a-j-(ia  +  d)  +  ia  +  2d')-\-(a-\-Sd)-{-.  .  .+a  +  (n—l)d  (1) 
^=/  +  (/_^)  +  (/-2^)  +  (/-3^)  +  .  .  .4./_(«_iy  (2) 

Adding  (1)  and  (2)  together  term  for  term,  noticing 
that  the  terms  containing  the  common  difference  niilif}- 
one  another,  we  have 

25=(«+/)4-(«4-/)  +  («+/)  +  («  +  /)  +  .  .  .  +  («+/). 
Since  the  number  of  terms  in  the  original  progression 
has  been  called  n,  we  write  the  last  equation 

2s=n(a  +  l), 
whence  the  formula  for  s, 

s=^n{a+l).  [2] 

396.  Formula  [1]  enables  us  to  obtain  the  value  of  / 
when  a,  n,  and  d  are  given,  or  the  value  of  «,  when  /,  u, 
and  d  are  given,  or  the  value  of  d  when  /,  a,  and  n  are 
given,  or  the  value  of  n  when  /,  a,  and  d  are  given.  Thus: 

(1)  Find  the  20th  term  of  3  + 8  +  13 -f-.   .   . 
Here  a=3,  d=5,  ^^=20,  therefore 

/=3  +  19x5=98. 

(2)  Find  the  number  of  terms  in  the  progresssion 
5  +  7  +  9  +  .   .  .  +  37. 

Here  a=5,  d=2,  1=37,  whence 

37=5  +  (;z-l)2. 
Solving  for  w,  n=17. 

(3)  Find  the  common  difference  in  a  progression  of  11 
terms  where  the  extremes  are  ^  and  30|-. 

Here  a=^,  /=30^  and  n=ll,  whence 

Solving  for  d,  d=3. 


ARITHIVIETICAL   PROGRESSIONS.  353 

(4)  Insert  3  arithmetical  taeans  between  5  and  21. 

Here  n=5,  <z=5,  and  /=21,  whence 
21  =  5  +  (5-iy. 
Solving  for  d,  d=A. 

Therefore,  the  means  are  9,  13,  and  17. 

397.  Formula  [2]  enables  us  to  find  any  one  of  the 

numbers  5,  n,  a  and  /,  when  the  value  of  the  other  three 
are  given.     Thus: 

(1)  Find  the  sum  of  10  terms  of  the  progression  where 
5  is  the  first  term  and  —58  the  last  term. 

Here  a=5,  7^=10,  /=— 58,  whence 
5=|-x  10(5-58.) 
That  is,  5=  —265. 

(2)  Find  the  number  of  terms  in  an  arithmetical  pro- 
gression where  the  first  term  is  4,  the  last  term  22,  and 
the  sum  91. 

Here  ^=4,  /=22  and  5=91,  whence 
91=4-^^C4  +  22.) 
Solving  for  n,  7i=l. 

398.  The  two  formulas 

J  l=a^(n-l)d  (1) 

t  s=\  n{a-^l)  (2) 

contain  five  different  letters,  hence  if  anj'-  two  of  them 
stand  for  unknown  numbers,  and  the  values  of  the  rest 
are  given,  the  value  of  the  two  unknown  numbers  can  be 
obtained  by  the  solution  of  a  system  of  two  equations. 
Thus: 

(1)  Find  the  sum  of  an  arithmetical  progression  where 
the  last  term  is  149,  the  common  difference  7,  and  t.^e 
number  of  terms  22. 

23 


354  PROGRESSIONS. 

Here  /=149,  d=l  and  ;^=22,  whence 

149=«  +  (22-l)7  (1) 

5=1x22(^  +  149.)  (2) 

From  (1),  a  =  1. 

Substituting  in  (2)  5=  1 661 . 

(2)  Find  the  first  term  of  an  arithmetical  progression 
of  21  terms,  whose  sum  is  1197  and  common  difference  is  4. 

Here  n=21,  5=  1197  and  ^=4,  whence 

/=^  +  (21-l)4  (1) 

1197=ix21(«+/).  (2) 

From  (1),  /=^  +  80.  (3) 

From  (2),  /=114-a.  (4) 

Whence,  /=97  and  ^=17. 

(3)  Find  the  number  of  terms  in  a  progression  whose 
sum  is  1095,  the  first  term  is  38  and  the  difference  is  5. 

Here  5=1095,  <2=38  and  d=b,  whence 

j         /=38-f(7z-l)5  (1) 

(  1095=i?2(38+/)  (2) 

From  (1),  /=33  +  5;z.  (3) 

From  (2),  2190=38;z  +  72/.  (4) 

Substituting  the  value  of  /  from  (3)  in  (4)  we  get 

2190=71;2+5«2.  (5) 

Solving  this  quadratic  equation,  we  find 
;^=15  or  -29.2. 
The  second  result  is  inadmissable,  since  the  number  of 
terms  cannot  be  either  negative  or  fractional. 

EXERCISE  138. 

Examples  and  Problems. 

Solve  each  of  the  following: 

1.  Given  <z=7,  ^=4,  n—\Z;  find /and  5. 

2.  Given  <2=^,  flf=6,  ;^=30;  find  /  and  s. 

3.  Given  «=9,  /=162,  ;2=52;  find  s  and  d. 


EXAMPLES    AND    PROBLEMS.  355 

4.  Given  a—hl,  /=80,  n—\%',  find  ^  and  d. 

5.  Given  /=242,  d?=21,  ;z=12;  find  «  and  s. 

6.  Given  /=1(5,  d——S,  ?i=dS]  find  ^  and  s. 

7.  Given  a  =  17,  /=3oO,  d?=9;  find  ;2  and  5. 

8.  Given  a=-2S,  /=28,  ^^=7;  find  ?i  and  ^. 

9.  Given  a=l^,  /=54,  ^=999;  find  7i  and  ^. 

10.  Given  «=5,  /=161,  ^=3320;  find  ;z  and  d. 

11.  Given  «=3,  «=50,  .y=3825;  find  /and  a?. 

12.  Given  ^=—45,  ?2=31,  5=0;  find  /and  ^. 

13.  Given  /=49,  n=19,  ^=503 J;  find  ^  and  d. 

14.  Given  /=10o,  ?i—16,  .9=840;  find  a  and  d. 

15.  Given  «=35,  j=2485,  ^=3;  find  «  and  /, 

16.  Given  «=25,  .?=— 25,  ^=J;  find  a  and  /. 

17.  Given  <.  =  4784,  a=41,  ^=2;  find  /and  ;2. 

18.  Given  .9=624,  a=d,  fl?=4;  find  /and  «. 

19.  Given  .y=278,  d=5,  /=77;  find  a  and  w. 

20.  Given  5=  1008,  ^=4,  /=88;  find  a  and  ^i. 

21.  What  is  the  sura  of  the  first  200  natural  numbers  ? 

22.  What  is  the  sum  of  the  even  numbers  from  0  to  200? 

23.  What  is  the  sum  of  the  odd  numbers  from  1  to  200? 

24.  What  is  the  sum  of  the  first  n  even  numbers? 

25.  What  is  the  sum  of  the  first  n  odd  numbers  ? 

26.  Insert  9  arithmetical  means  between  —^  and  -|-|^. 

27.  Sum  the  series  v'''f+l/2  +  3l/|+.  .  .  to  20  terms. 

28.  Sum  the  series  5—2—9—.  .  .  to  8  terms. 

29.  Insert  5  arithmetical  means  between  10  and  8. 

30.  Insert  4  arithmetical  means  between  —2  and  —16. 

31.  Sum  («  +  ^)2  +  (a2_|_^2)_j_(^_^)2  to  n  terms. 

32.  Find  the  sum  of  the  first  10  multiples  of  3. 


356  PROGRESSIONS. 

33.  Find  the  sum  of  the  first  50  multiples  of  7. 

34.  Find  the  sum  of  the  odd  numbers  between  200 
and  300. 

35.  Find  the  arithmetical  progression  whose  sum  is 
550  and  whose  middle  term  equals  50. 

36.  The  sum  of  25  successive  terms  of  the  progression 
5  +  8  +  11  +  .  .  .  is  1025;  what  is  the  first  term? 

37.  The  sum  of  10  terms  of  an  arithmetical  progression 
is  15,  and  the  fifth  term  is  0  ;  what  is  the  first  term  ? 

38.  How  many  terms  of  the  progression  9  + 13  +  17-f .  . 
must  be  taken  in  order  that  the  sum  may  equal  624  ? 

39.  We  must  take  how  many  terms  of  the  progression 
(^l-f--W(l+-T)+(l+-T)+  .  .  in  order  that  the  sum 
may  be  6A  ? 

40.  How  many  terms  must  be  taken  from  the  com- 
mencement of  the  series  1  +  5  +  94-13  +  17.  etc.,  so  that 
the  sum  of  the  13  succeeding  terms  shall  be  741  ? 

41.  The  sum  of  the  first  three  terms  of  an  arithmetical 
progression  is  15,  and  the  sum  of  their  squares  is  83  ; 
find  the  common  difference. 

Let  ;c=first  term  and/  the  common  difference.     Then 
X  +  {x  +y)  +  {x  +  2y)=  15, 
and  A:2  +  (x+/)3f(a;  +  2y)2=83. 

Another  notation  which  is  very  convenient  in  a  problem  like  this 
is:  Represent  the  three  terms  by  x—y,  x,  and  x+y,  whence  we 
would  write  i.x—y)   -i-x  +{x-j-y)  =15, 

and  (ar— 7)2-fa;2-{-(x-|-,y)3  =  83. 

42.  There  are  two  arithmetical  progressions  which 
have  the  same  common  difference  ;  the  first  terms  are 
3  and  5  respectively,  and  the  sum  of  seven  terms  of  the 
the  one  is  to  the  sum  of  seven  terms  of  the  other  as  2  to  3. 
Determine  the  progressions. 


GEOMETICAI.    PROGRESSIONS.  357 

43.  The  sura  of  three  numbers  in  arithmetical  pro- 
gression is  12,  and  the  sum  of  their  vSquares  is  66.  Find 
the  numbers. 

44.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  33,  and  the  sum  of  their  squares  is  461.  What 
are  the  numbers? 

EXERCISE  139. 

Geometrical  Progressions. 

399.  A  Geometrical  Progression  is  a  series  of  terms 
such  that  each  term  is  the  product  of  the  preceding  term 
by  a  fixed  factor  called  the  Ratio.  The  following  are 
examples : 

(1)  3,  6,  12,  24,  48.  (3)  ^,  i,  i,  ^V,  ^V 

(2)  100,  -50,  25,  -121-.      (4)  a,  ar,  ar"- ,  ar^,  ar\ 
The  first  and  last  terms  are  often  called  the  Extremes 

and  the  other  terms  the  Means. 

400.  The  71  th  Term  of  a  Geometrical  Progression. 

Let  a  represent  the  first  term  of  any  geometrical  progres- 
sion, and  r  the  ratio.  Then  the  progression  may  be  written 

No.  of  term:     1.      2.       3.         4.         5. 

Progression:  «,  ar,  ar"^,  ar^,  ar^,  .  .  . 
We  notice  that,  by  the  nature  of  the  progression,  every 
time  the  number  of  terms  is  increased  by  1  the  exponent 
of  r  is  increased  by  1  also,  and  the  exponent  of  r  in  any 
term  is  one  less  than  the  number  of  that  term.  Therefore, 
representing  the  n  th  term  by  /, 

l=ar"-^.  [3] 

401.  The  Sum  of  ?t  Terms  of  a  Geometrical  Pro- 
gression. Representing  by  ^  the  sum  of  n  terms  of  any 
geometrical  progression,  we  have 

s=:^a-\-ar-{-ar^  +ar^  -{- .  .  .-^ar"-""  ■i-ar"-\  (1) 


35^  PROGRESSIONS. 

Multiplying  this  equation  by  r,  we  get 

rs—ar+ar^  +  ar^  +  ar"^-^.  .  .-{-ar"~'^+ar'\         (2) 
Subtracting  (1)  from  (2),  we  have 

rs — s=ar"—a.  (3) 

Whence  s(r—l')=ar"—a, 

or  8= —  [4] 

Now,  from  [3]  l=ar"~^.  Therefore,  ar"=r(ar"-^)==r/, 
and  [4]  may  be  written 

rl  —  a  TM 

S= -. —  [5] 

402.  We  give  a  few  examples  of  the  use  of  formulas 
[3],  [4],  and  [5]. 

(1)  Find  the  7th  term  of  the  progression  4  +  8  +  16  +  .  . 
Here  «=4,  r=2,  and  7i=l ,  whence 

/=4x2«==256. 

(2)  Find  sum  of  6  terras  of  progression  13+1. 3  +  . 13  + 
Here  ^=13,  «=6,  and  ?"=xV>  whence 

_13xGV)«-13 

_    13-13000000    _12999987_  . 

that  IS,     "^- 100000- lOOOOOO"    900000  -^^•4444o. 

(3)  Insert  3  geometrical  means  between  31  and  496. 
Here  a=31,  /=496,  and  ?2=5,  whence 

496  =  31x^4, 
or  r*  =  16, 

therefore,  r==fc2. 

Consequently,  the  required  means  are  62,  124,  and  248, 
or  -62,  +124,  and  —248. 

403.  The  two  equations 

_ar"—a 
r—1 


GEOxMETICAL    PROGRESSIONS.  359 

contain  five  letters.  If  any  two  of  them  are  unknown 
numbers  and  the  values  of  the  other  three  are  given,  the 
value  of  the  two  unknown  numbers  can  be  determined 
b}^  solving  the  system  of  two  equations.  But  if  r  is  an 
unknown  number,  the  equations  of  the  system  are  of  a 
high  degree,  since  n  is  usually  a  large  number  and 
always  greater  than  2  at  least.  In  this  case  we  will  be 
unable  to  solve  the  system,  as  it  is  beyond  the  range  of 
Chapter  XV.  Also,  if  n  is  an  unknown  number,  we  will 
have  an  equation  with  the  unknown  number  appearing 
as  an  exponent,  which  is  a  kind  of  equation  we  have  not 
yet  considered.  Hence  there  are  a  limited  number  of 
cases  in  which,  with  our  present  means,  we  can  solve 
the  above  system.  We  give  a  few  examples  of  the  cases 
readily  solved. 

(1)   Find  the  sum  of  a  geometrical  progression  of  7 
terms,  of  which  the  last  term  is  128,  the  ratio  being  2. 
Here  /=r28,  r=2,  and  w=7,  whence 

(  128=a26  '  (1) 


(        '-       1 
From  (1)  ^  =  2,  whence,  from  (2),  5=  254. 


(2) 


(2)  Find  the  sum  of  a  geometrical  progression  of  5 
terms,  the  extremes  being  8  and  10368. 

Here  «=8,  /=10oG8,  and  ;^  =  5,  whence 

(  103G8=8r^  (1) 

\        r  10368 -8 

From  (1)  r=6,  whence  from  (2)  ^=12450. 

(3)  Find  the  extremes  of  a  geometrical  progression 
whose  sum  is  635,  if  the  ratio  is  2  and  the  number  of 
terms  7. 


36o  PROGRESSIONS. 

Here  ^=635,  r=2,  and  n=l ,  whence 

(/=^26  (1) 

j  635=?^  (2) 

Substituting  /from  (1)  in  (2),  we  get 

635=128«— «. 
Whence  a=5  ;  hence  /=320. 

(4)  The  4th  term  of  a  geometrical  progression  is  4,  and 
the  6th  term  is  1.     What  is  the  10th  term? 

Here  «r^=4  (1) 

and  ar'^  =  l  (2) 

Whence,  by  dividing  (2)  by  (1), 

^2 1 

Whence,  r=d=|-. 

4 
Therefore,  from  (1),         ^=--=±32. 

Then  the  10th  term  is  ±32(=bi)9=J^. 

EXERCISE  140. 

Examples  and  Problems. 

1.  Find  the  sum  of  7  terms  of  4+8+ 16+.  . 

2.  Find  the  sum  of  9  terms  of  2 +  6  + 18  +  .  . 

3.  Find  the  sum  of  7  terms  ofl+4+16  +  .  . 

4.  Find  the  sum  of  11  terms  of  9  +  3  +  1  +  .  . 

5.  Find  the  sum  of  10  terms  of  l  +  i+i+,  . 

6.  Find  the  10th  term  and  the  sum  of  10  terms  of 
4-2  +  1-.  .  . 

7.  Sum  the  series  1^3  +  1^^6+1^12  +  .  .  .  to  8  terms. 

8.  Sum  the  series  3— 2+|— f +.  .  .  to  9  terms. 

g.  Sum  the  series  —4+8—16+32—.  .  .  to  6  terms. 
lo.  Sum  «  +  «(l+;r)  +  «Cl+^)2  +  .  .  .  to  8  terms. 


EXAMPLES    AND    PROBLEMS.  36 1 

11.  Sum  a-\ — 5^  +  7-^ — ?T2  +  -  •  •  to  10  terms. 

12.  Sum  d(l-j-xy-^+d(l-}-xy-'^-{-.  .  .  to  ?2  terms. 

13.  Snmx"-^+x"-^y+x"-^y'^-\-x"-^y^  +  .  .  to  ;e  terms. 

14.  Snmx"-'^—x"-y-j-x"-^y'^—x"-*y^  +  .  .  to  w  terms. 

15.  Find  r  and  s;  given  a=2,  /=31250,  ^=7. 

16.  Find  rand  s;  given  <2=36,  /=^,  n=7.        ; 

17.  Find  rand  s;  given  a=S,  /=49152,  n=S. 

18.  Find  rand  s;  given  a=7,  /=3584,  72=10. 

19.  Insert  2  geometrical  means  between  47  and  1269. 

20.  Insert  3  geometrical  means  between  2  and  3. 

21.  Insert  1  geometrical  mean  between  14  and  686. 

22.  Given  /=78125,  r=5,  n=S  ;  find  a  and  s. 

23.  Given  /=^V,  ^=i,  7z=5  ;  find  a  and  j. 

24.  Given  5=635,  72=7,  r=2  ;  find  a  and  /. 

25.  Select  6  terms  from  the  progression  •^—2  +  8—.  .  . 
whose  sum  shall  equal  —6536. 

26.  The  sum  of  the  extremes  of  a  geometrical  pro- 
gression of  4  terms  is  56,  and  the  sum  of  the  means 
is  24.     Find  the  4  terms. 

27.  Insert  7  geometrical  means  between  a^  and  d^. 

28.  Given  a—^,  /=1024,  ;2=14  ;  find  r  and  s. 

29.  Sum  the  series  2+22+P  +  2^"^2^"^26"^'  '  '  ^^  ^ 

'  /I     ,      3      ,      ^  \    ,    /"l  ^    ^   _!_   ^  \  1    _L 

terms,  or  the  progression  1 2+22  "^2^/     \2     2^  '^2^/2^'^ 
to  3  terms. 

30.  Find  the  sum  of  the  first  10  consecutive  powers  of  2. 

31.  Find  sum  of  the  first  10  consecutive  powers  of  —  |. 

32.  Sum   the  progression  .272727  ...   or  TVTr+rffVTnr 
+  Towtnrir+-  •  •  to  6  terms. 


362  PROGRESSIONS. 

33.  Sum  a—ar~^-\-ar~'^—ar~^-{-.  .  .  to  n  terms. 

34.  The  4th  term  of  a  geometrical  progression  is  192 
and  the  7th  term  is  12288  ;  find  the  sum  of  the  first  3 
terms. 

35.  The  6th  term  of  a  geometrical  progression  is  150, 
and  the  8th  term  is  7644  ;  what  is  the  4th  term  ? 

36.  Prove  that  if  numbers  are  in  geometrical  progres- 
sion their  differences  are  also  in  geometrical  progression, 
having  the  same  common  ratio  as  before. 

37.  If  a  +  d-\-c-{-d+.  .  .  is  a  geometrical  progression, 
prove  that  (a^  +  d'')-i-(id^  +c'-)  +  {c'^  -i-d^)  + .  .  .  is  also  a 
geometrical  progression. 

38.  A  man  agreed  to  pay  for  the  shoeing  of  his  horse 
as  follows  :  .0001  cents  for  the  first  nail,  .0002  cents  for 
the  second  nail,  .0004  cents  for  the  third  nail,  and  so  on 
until  the  8  nails  in  each  shoe  were  paid  for.  How  many 
dollars  did  he  agree  to  pay  ?  How  much  did  the  last 
nail  cost  him  ? 


CHAPTER  XX. 

BINOMIAL  THEOREM. 

EXERCISE  141. 

The  Lav/s  of  Exponents  and  Coefficients. 

404.  The  Binomial  Theorem  enables  us  to  find  any 
power  of  a  binomial  without  the  labor  of  obtaining  the 
previous  powers;  in  other  words,  it  enables  us  to  obtain 
any  power  of  a  binomial  without  actually  performing  the 
multiplication. 

405.  Let  us  obtain  several  powers  of  x-\-a  by  actual 
multiplication  : 

\st  power ^     x+a 


2d  poiver^ 


Sd  power, 


4itk  pozver, 


x+a 
x'^+  ax 

ax-{-a" 

x'^-\-2ax+a^ 

x+a 

x'^  +  2ax'^+  a-x 

ax'^+2a-x-{-x^ 

x^+Sax^  +  Sa^-x+x^ 
x+a 

x^  +  Sax'^  +  Sa^-x''+  a'^x 

ax^+^a^-x^+Za'^x+a^ 

x'^  +  4ax^  +  i]a^-x"-+4a'^x+a^ 
x+a 

x^+Aax^+   {5a'^x^+   Aa^x'^ 
ax^+  Aa-x^+  6a^x- 

'+  a^x 
'+4a^x+a^ 

hth  power,     x^  +  6ax'^  +  10a^-x'^  +  10a'Kr^  +da'^x+a^ 

When  any  power  of  x  +  a  is  written  out  in  full,  as  in  the 
above,  it  is  called  the  Expansion  of  that  power  of  r:  +  a. 


364  BINOMIAL  THEOREM. 

406.  The  Law  of  Exponents.  We  notice  in  the 
expansion  of  the  different  powers  of  x+a  that  x  appears 
in  each  term  except  the  last,  and  that  a  appears  in  each 
term  except  the  first,  and  that  both  x  and  a  occur  in 
each  of  the  other  terms.     We  observe  also  that 

THE  EXPONENTS  OF  X  FOLLOW      THE  EXPONENTS  OF  a  FOLLOW 
THE  FOLLOWING  SCHEME.  THE  FOLLOWING  SCHEME. 

In  (x+ay         10  0  1 

In  (x+ay         2  10  0  12 

In  (x+ay         3210  0123 

In  (x+ay         43210  01234 

In  (_x+ay         543210  012345 

It  is  a  necessary  consequence  of  the  successive  multi- 
plication by  x+a,  that  the  exponents  v/ill  continue  to 
fall  into  the  above  schemes,  for  at  each  multiplication  all 
the  power  of  x,  likewise  of  a,  are  increased  by  1 ,  and 
there  will  alwa3^s  be  one  term  which  does  not  contain  x, 
and  always  one  which  does  not  contain  a ;  therefore  the 
law  of  exponents  : 

I7Z  any  power  of  a  binomial,  x-\-a,  the  exponent  of  x 
begins  in  the  first  term  with  the  exponent  of  the  poiver,  afid 
in  the  following  terms  co7itinually  decreases  by  07ie.  The 
exponent  of  a  com?}tences  with  one  in  the  second  term,  and 
c'07itinually  i7icreases  by  one. 

407.  Number  of  Terms.  We  observe  that  in  the 
first  power  there  are  two  terms,  in  the  second  power  there 
are  three  terms,  in  the  third  power  there  are  four  terms, 
and  so  on.  In  any  power  of  x-\-a  there  is  a  term  con- 
taining each  of  the  powers  of  x,  as  high  as  the  required 
power  of  ^4-^,  and  one  term  which  does  not  contain  x  \ 
therefore, 

The  7izt77iber  of  terms  i7i  any  power  of  a  bi7iomial  is 
always  one  greater  than  the  ext>07ient  of  the  poiver. 


LAWS  OF  COEFFICIENTS  AND  EXPONENTS.       365 

408.  The  Law  of  Coefficients.  First  Statement. 
The  coefi&cients*  in  any  power  of  x+a  are  of  course  the 
sums  of  the  coefficients  in  the  two  partial  products  which 
are  added  together  to  produce  the  required  power.  Now 
the  coefficients  in  each  of  the  partial  products  are  just 
alike,  and  each  partial  product  has  the  same  coefficients 
as  the  multiplicand,  or  next  lower  power  of  x-\-a  to  the 
one  we  are  finding.  Thus  the  coefficients  which  are 
added  together  to  produce  the  coefficients  in  (^x-\-cC)^ 
are  as  follows : 

18    3     1 
13    3     1 

14     6     4     1 

The  second  coefficient  in   (x-\-aY  is  the  sum  of  the 
second  and  Jirst  coefficients  in  {x+a)^,  the  t/ii'rd  coefficient 
in  (x+a)*  is  the  sum  of  the  t/ii'rd  and  second  coefficients 
in  {x-\-cC)^,  i\iQ  fourth  coefficient  in  {x-\-d)^  is  the  sum 
oi  Xho:  fourth  and  third  coefficients  in  (x-\-aY,  th&  fifth 
coefficient  in  {x-\-d)^  is  the  sum  of  the  fifth  (which  is  0) 
and  fourth  coefficients  in  (x-^a)''.     Writing  down  the 
coefficients  in  the  different  powers  of  :r+a,  we  may  present 
this  same  truth  in  a  little  different  form  : 
Coefficients  in  {x-\-aY     1     1 
Coefficients  in  {x-\-ay     12     1 
Coefficients  in  {x+aY     13     3     1 
Coefficients  in  \x-\-ay     14     6     4     1 
Coefficients  in  {x+aY     1     5  10  10     5     1 
By  arranging    the    coefficients    in    this    triangle   we 
may  say  that  each  coefficient  is  the  S2im  of  the  coefficient 
immediately  above  it  aiid  the  coefficient  imtnediately  to  the 
left  of  this  last. 

*  By  the  coefficients  in  any  power  of  ;r+a  is  meant  the  coefficients  of  the  powers 
ofx  and.  a.     This  lan.ernage  is  not  exact,  (see  Art.  14.)  but  it  is  c<:stomary. 


366  BINOMIAL  THEOREM. 

Since  the  partial  products  will  continue  to  be  combined 

in  the  way  we  have  noticed  if  we  continue  to  multiply 

by  x-\-a,  we  may  extend  the  above  triangle  indefinitely 

and  get  the  coefficients  in  any  power  oi  jt+«  we  wish: 

Coefficients  in  (x-{-a)'^    1  1 


Coefficients  in  (x-\-ay^ 

1  2     1 

Coefficients  in  {x-\-ay 

13     3     1 

Coefficients  in  {x+ay 

14     6     4     1 

Coefficients  in  {x+ay 

1  5  10  10     5     1 

Coefficients  in  {x-^a)^ 

1  6  15  20  15     6 

1 

Coefficients  in  {x-\-ay 

1  7  21  35  35  21 

7  1 

Coefficients  in  {x-\-ay 

1  8  28  56  70  56 

28  8 

409.  The  Law  of  Coefficients.  Second  Statement. 
It  is  more  usual  to  use  a  law  of  coefficients  different  from 
the  one  given  in  the  previous  article.  It  is  found  that 
fhe  coefficient  in  any  term  in  a  power  of  a  binomial  can  be 
made  from  the  coefficient  and  the  two  exponents  in  the  pre- 
ceding term.  Thus  consider  "Oa^  fourth  power  oi  x-\-a, 
x^-h4ax^-hQa'^x'^  -h4a^x-^a^. 

The  first  coefficient  is  1,  and  the  second  coefficient  is  4, 
the  exponent  of  the  power;  the  t^ird  coefficient,  6,  is  the 
preceding  coefficient,  4,  multiplied  by  3,  the  exponent 
of  X  in  the  second  term,  divided  by  2,  one  more  than  the 
exponent  of  a  in  the  second  term.     Also,  4,  the  fourth 

6x2 
coefficient,  is  — o~>  where  6  is  the  coefficient  in  the  pre- 
ceding term,  2  is  the  exponent  of  x,  and  3  is  one  more 
than  the  exponent  of  a;  and  so  on. 
lyikewise  in  the  fifth  power  of  x-\-a, 

x^  +  bax^-]-lOa'^x'^  +  10a^x^-\-bax^+x^, 
the  coefficient  in  the  first  term  is  1 ;  the  coefficient  in  the 
second  term  is  5,  the  exponent  of  the  required  power ; 
the  coefficient  in  the  third  term,  10,  is  the  coefficient  in 


LAWS  OF  COEFFICIENTS  AND  EXPONEN  IS.        367 

the  preceding  term,   5,   multiplied  by  4,   the  exponent 
of  X,  and  divided  by  2,  one  more  than  the  exponent  of  a; 

10,  the  coefficient  in  the  fourth  term,  is  similarly  — ^-v-; 

o 

10x2 
5,  the  next  coefficient,  is  — ^ — J  ^"^  1»  the  last  coefficient, 

is   -TT    .     The  next  coefficient  would  be  —77-. 
0  b 

After  treating  any  other  of  the  first  five  powers  in  the 
same  way,  we  would  find, 

In  any  of  the  first  Jive  powers  of  x-\-a,  the  coefficient  in 
the  first  term  is  1,  that  in  the  second  term  is  the  expo7ient  of 
the  power y  and  if  the  coefficient  in  any  term  be  multiplied  by 
the  exponent  of  x  in  that  term  and  divided  by  the  exponent 
of  a  increased  by  one,  it  will  give  the  coefficient  in  the  suc- 
ceeding term. 

410.  The  law  stated  in  the  last  article  may  be  observed 
to  be  true  in,  any  of  the  five  powers  of  Jt-f-^  that  we  have 
actually  worked  out ;  it  now  remains  to  prove  that  the 
same  law  holds  for  ^//powers  oi x-\-a. 

lyCt  71  stand  for  a  positive  whole  number,  and  suppose 
we  wish  to  find  {x-\-d)''.  We  know  from  Art.  406  that 
the  terms  without  the  coefficients  will  be  as  follows  : 

x*\  ax"-\  a'^x"--,  a^x"-^,  a^jtr"-*,  etc. 
Now,  if  the  law  stated  in  the  previous  article  does  hold, 
we  would  write  (x-{-a)"= 

i^this  is  true,  by  Art.  408  the  coefficients  in  (x-tay^^  are 
n(7i-l).        ;^(;^-l)(7^-2)     n(n-l) 

1,  ^  +  1.  2 ^    '  2xS 2 ' 

n(n-lXn-2Xn-S)     n(?i-l)(7t-2) 
2x3"x4  "^  2x3 


368  BINOMIAL  THEOREM. 

or,  removing  a  common  factor  from  the  third,  fourth, 
etc.,  of  these  expressions,  we  would  write  them 

nCn-lXn-2){^^+^),  etc. 
or,  performing  the  additions  in  the  large  parentheses, 
1,  n  +  1,   — ^-,  2^ , 

(?g+l)^gOg— 1)0?— 2) 

2x"3x4  '  ^^^• 

or,  supplying  the  powers  of  j«;and  a  for  the  (n+1)  power 
of  x+a  by  Art.  406,  we  get 

Z  Z  "K  o 

+  2^3x4  «  -^      +■  •  •         (^) 

Now  we  know  that  equation  (2)  is  true  if  equation  (1) 
IS  i7'ue.  But  equation  (2)  is  of  exactly  the  same  form 
as  (1),  merely  having  {n-{-V)  in  place  of  n,  each  coeffi- 
cient being  obtainable  from  the  coefficient  and  exponents 
of  the  preceding  term  by  the  law  of  Art  409.  Therefore 
we  have  proved  that  the  law  of  coefficients  of  Art.  409  holds 
in  the  {7i-\-V)  power  of  x-\-a  if  it  holds  in  the  nth  power 
of  x-\-a.  But  we  know  this  law  of  coefficients  holds  in  the 
5th  power  of  x-\-a  ;  therefore  it  follows  that  it  holds  in 
the  6th  power.  Now  we  know  this  law  of  coefficients 
holds  in  the  6th  power  of  x-j-a,  and  therefore  it  holds  in 
the  7th  power;  therefore  it  holds  in  the  8th  power,  and  so 
on.     Therefore  it  holds  universally. 

Thus  we  have  proven  the  law  of  Art.  409  holds  for 
any  power  of  a  binonial. 


LAWS  OF  COEFFICIENTS  AND  EXPONENTS.       369- 

412.  The  Statement  of  the  laws  of  exponents  and 
coefficients  for  any  power  of  x-\-a  is  called  the  Bino- 
mial Theorem,  and  is  usually  given  as  follows  : 

I.  Exponents.  In  any  poivcr  of  a  binomial,  x+a,  ihe 
exponent  of  x  begins  in  the  /irst  term  with  the  exponent  of 
the  power,  a7id  in  the  folloiving  terms  continually  decreases 
by  one.  The  exponent  of  a  commences  with  07te  in  the  sec- 
ond term  of  the  power,  a)id  continually  increases  by  07ie; 

II.  CoKFFiciENTS.  The  coefficient  iii  the  first  term  is 
one,  that  in  the  second  term  is  the  exponent  of  the  power;- 
and  if  the  coefficient  in  any  term  be  multiplied  by  the  expo- 
nent of  X  in  that  term  and  divided  by  the  exponent  of  a 
increased  by  one,  it  will  give  the  coefficient  in  the  succeeding 
term. 

413.  The  expansion  of  (jr=b«)"  is  usually  called  the 
Binomial  Formula. 

If  in  equation  (1),  Art.  400,  we  substitute  ±a  for  a,  we 

get  the  following  as  the  expansion  of  (jrdra)": 

[x±a)"= 

.,  ,  ,  n{n—\)   o    „  ,  .  n{n—\)[n—2)   ,    .,  , 
ac"  ±  nax"-^  -\ — -a-x"--  ±  — — ^^ -a'^x"-^ 

+"-^r^^«*--* . . .  t.i 

Therefore,  in  any  power  of  the  difference  of  two  num- 
bers the  sign  of  the  first  term  is  +,  of  the  second  —,  and 
so  on,  alternately  +  and  — . 

414.  We  will  now  give  a  few  examples  of  the  use  of 
the  binomial  theorem. 

(1)  Expand  {a^-by. 

We  may  expand  this  at  once  by  the  theorem  as  follows: 
flC+6a5^+15«4^2^20«^^3  +  15^-/^^  +  6^/^^  +  <^6^ 

or  we  may  substitute  x^a,  a—b,  and  «=G  in  formula  [1] 
21 


370  BINOMIAL  THEOREM. 

.0x5x4x3x2  ,.,  0x5x4x3x2x1,, 


2x3x4x5     "^  2x3x4x5xG 

which  reduces  to  the  same  result  as  before. 

(2)  Expand  (z^  +  3_y)^ 

Here  x=il  and  a='^y.     By  the  theorem  we  get 

Performing  the  indicated  operations,  we  get 

?r^  +  15?^V+^0^^V^ +  270/^2^3  _|_405?/;/'*-f243j'.5. 

(3)  Expand  (^^-2)4. 

Here  x^r"- ,  a=—2,  and  ;^=4.     By  the  theorem 

Performing  the  indicated  operations,  we  get 

5K8_8r6  +  24r-*-32r2^1G. 
.   (4)  Expand  (2^-iy. 
Here  x=3^,  «=— |,  7^=3.     By  the  theorem, 
(3^)^-3(3<^)2(i)4-3(3^)(i)2-(i)«. 
Performing  the  indicated  operations,  we  get 
27^^-V^-+|^-i. 

EXERCISE  142. 

Examples. 

Expand  each  of  the  following  by  the  binomial  theorem 
or  formula  : 

1.  {a+xy\  5.   {\^ay\  9.   {m'+zcy, 

2.  {b+xy.  6.  (2+xy,  10.   {a-xy, 

3.  {d-\-yy,  7.   (2-A-)*.  II.   (5flr-3;0^ 

4.  (^+;r)«.  8.  (i+ji-)^  12.   (a^-b'^y. 


EXAMPLES.  371 

13.  i-x+2ay.  17.   (d'^-c^-)^.  21.  (2/i''-Bx-^y. 

14.  (2x+Say.  18.   Gr+2r)*,  22.  (3;i;2_l)4. 

15.  (1-xy.  19.   (Sa+iy.  23.  (T/«+;r)«. 

16.  (1— rt)8.  20.   (2^zjir-.r2)4.  24.  (2^-i/i)5 

25-   G'Tj'— :«:-)^  31.   («— <^+:r— 2)3. 


2        3. 


26.  (]/^^— l?^rt/^)».         32.   (jtra+.r^)^ 

27.  (^  +  [:r+;/])3.  33.    («-2_^i)4^ 

c8.   (la-i-d]-2y.  34.   (.r2  4-2«j»:+rt2)3. 

29.  («+^-ji')'.  35.  {y'c''-2xy\ 

30.  ([a4-/^]  +  [^+^])^  36.  {^-^■^)'- 

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